| L(s) = 1 | − 2-s + 4-s + 8·5-s + 22·7-s + 15·8-s − 8·10-s + 28·11-s + 30·13-s − 22·14-s − 31·16-s + 51·17-s + 80·19-s + 8·20-s − 28·22-s − 142·23-s − 267·25-s − 30·26-s + 22·28-s + 456·29-s + 230·31-s − 17·32-s − 51·34-s + 176·35-s + 356·37-s − 80·38-s + 120·40-s + 294·41-s + ⋯ |
| L(s) = 1 | − 0.353·2-s + 1/8·4-s + 0.715·5-s + 1.18·7-s + 0.662·8-s − 0.252·10-s + 0.767·11-s + 0.640·13-s − 0.419·14-s − 0.484·16-s + 0.727·17-s + 0.965·19-s + 0.0894·20-s − 0.271·22-s − 1.28·23-s − 2.13·25-s − 0.226·26-s + 0.148·28-s + 2.91·29-s + 1.33·31-s − 0.0939·32-s − 0.257·34-s + 0.849·35-s + 1.58·37-s − 0.341·38-s + 0.474·40-s + 1.11·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3581577 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3581577 ^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.798831084\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.798831084\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 17 | $C_1$ | \( ( 1 - p T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 + T - p^{4} T^{3} + p^{6} T^{5} + p^{9} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - 8 T + 331 T^{2} - 2032 T^{3} + 331 p^{3} T^{4} - 8 p^{6} T^{5} + p^{9} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 22 T + 891 T^{2} - 14300 T^{3} + 891 p^{3} T^{4} - 22 p^{6} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 28 T + 2627 T^{2} - 69844 T^{3} + 2627 p^{3} T^{4} - 28 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 30 T + 5119 T^{2} - 141212 T^{3} + 5119 p^{3} T^{4} - 30 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 80 T + 15945 T^{2} - 757312 T^{3} + 15945 p^{3} T^{4} - 80 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 142 T + 20731 T^{2} + 1854884 T^{3} + 20731 p^{3} T^{4} + 142 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 456 T + 127075 T^{2} - 23761392 T^{3} + 127075 p^{3} T^{4} - 456 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 230 T + 77787 T^{2} - 13785468 T^{3} + 77787 p^{3} T^{4} - 230 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 356 T + 133995 T^{2} - 29888184 T^{3} + 133995 p^{3} T^{4} - 356 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 294 T + 120199 T^{2} - 38886804 T^{3} + 120199 p^{3} T^{4} - 294 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 556 T + 289617 T^{2} - 81141512 T^{3} + 289617 p^{3} T^{4} - 556 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 640 T + 396797 T^{2} + 134564608 T^{3} + 396797 p^{3} T^{4} + 640 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 302 T + 293171 T^{2} + 71759636 T^{3} + 293171 p^{3} T^{4} + 302 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 636 T + 514369 T^{2} + 211823016 T^{3} + 514369 p^{3} T^{4} + 636 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 84 T + 556531 T^{2} + 31340024 T^{3} + 556531 p^{3} T^{4} + 84 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 1008 T + 967329 T^{2} - 607104160 T^{3} + 967329 p^{3} T^{4} - 1008 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 402 T + 483859 T^{2} - 12894428 T^{3} + 483859 p^{3} T^{4} - 402 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 838 T + 1394903 T^{2} - 671950004 T^{3} + 1394903 p^{3} T^{4} - 838 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 594 T + 357843 T^{2} - 156405492 T^{3} + 357843 p^{3} T^{4} + 594 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 2396 T + 3204249 T^{2} - 2882084008 T^{3} + 3204249 p^{3} T^{4} - 2396 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 170 T + 1042603 T^{2} + 206881916 T^{3} + 1042603 p^{3} T^{4} - 170 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 270 T + 2151919 T^{2} + 286220420 T^{3} + 2151919 p^{3} T^{4} + 270 p^{6} T^{5} + p^{9} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.17733506649139507468506665512, −11.02737283016101839651636562940, −10.31819639495361563011258468862, −10.17207865755284213327310921791, −9.624396572606530069461395954809, −9.526513657271677198750921965248, −9.415087799851179664143929882773, −8.396212568951396220914393036712, −8.384103942534636298713570385177, −8.002435324201369610023989484499, −7.66339150973919625829796199995, −7.52411789311406541898030271039, −6.65283483138256497126441004186, −6.33393508384993200101546082744, −5.96520668290534900983468579294, −5.87803139257014233075472377117, −4.92366682842881873154332931004, −4.64229333094379753346661223765, −4.47662001400708767625410092592, −3.66187611369757625938922168409, −3.25105324535129063392862491881, −2.30137169809090431660128705933, −1.98678942895908199373291215876, −1.19248705018502435910495260381, −0.888174854581830993379655277345,
0.888174854581830993379655277345, 1.19248705018502435910495260381, 1.98678942895908199373291215876, 2.30137169809090431660128705933, 3.25105324535129063392862491881, 3.66187611369757625938922168409, 4.47662001400708767625410092592, 4.64229333094379753346661223765, 4.92366682842881873154332931004, 5.87803139257014233075472377117, 5.96520668290534900983468579294, 6.33393508384993200101546082744, 6.65283483138256497126441004186, 7.52411789311406541898030271039, 7.66339150973919625829796199995, 8.002435324201369610023989484499, 8.384103942534636298713570385177, 8.396212568951396220914393036712, 9.415087799851179664143929882773, 9.526513657271677198750921965248, 9.624396572606530069461395954809, 10.17207865755284213327310921791, 10.31819639495361563011258468862, 11.02737283016101839651636562940, 11.17733506649139507468506665512
