| L(s) = 1 | + 4·2-s + 12·4-s − 14·5-s + 14·7-s + 32·8-s − 56·10-s + 22·11-s − 104·13-s + 56·14-s + 80·16-s + 22·17-s + 2·19-s − 168·20-s + 88·22-s + 40·23-s + 10·25-s − 416·26-s + 168·28-s + 96·29-s + 238·31-s + 192·32-s + 88·34-s − 196·35-s + 392·37-s + 8·38-s − 448·40-s + 110·41-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 1.25·5-s + 0.755·7-s + 1.41·8-s − 1.77·10-s + 0.603·11-s − 2.21·13-s + 1.06·14-s + 5/4·16-s + 0.313·17-s + 0.0241·19-s − 1.87·20-s + 0.852·22-s + 0.362·23-s + 2/25·25-s − 3.13·26-s + 1.13·28-s + 0.614·29-s + 1.37·31-s + 1.06·32-s + 0.443·34-s − 0.946·35-s + 1.74·37-s + 0.0341·38-s − 1.77·40-s + 0.419·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(8.383475537\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.383475537\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 14 T + 186 T^{2} + 14 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 8 p T + 6646 T^{2} + 8 p^{4} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 22 T + 7122 T^{2} - 22 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 46 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 40 T + 8462 T^{2} - 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 96 T + 28934 T^{2} - 96 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 238 T + 64590 T^{2} - 238 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 392 T + 128422 T^{2} - 392 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 110 T + 70242 T^{2} - 110 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 288 T + 134550 T^{2} + 288 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 826 T + 377198 T^{2} - 826 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 472 T + 298758 T^{2} - 472 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 100 T - 140442 T^{2} + 100 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 504 T + 354294 T^{2} + 504 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 320 T - 170202 T^{2} - 320 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 908 T + 758766 T^{2} - 908 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 1738 T + 1527658 T^{2} + 1738 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 984 T + 821342 T^{2} - 984 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 242 T + 1144542 T^{2} - 242 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 2884 T + 3473030 T^{2} - 2884 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 628 T + 1048870 T^{2} - 628 p^{3} T^{3} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.343616970536836988631170542202, −9.112554945344679852923605407829, −8.315035302192253340514995968085, −7.990580720826737929644451502840, −7.71679793230319959346671483502, −7.38616459036496976541630735904, −6.82003096090968680588406798545, −6.75436469070911065335904160188, −5.83024493637957320724794753856, −5.70354752266600993680057293765, −5.04542553271678736060633061044, −4.56481769788834067678814234164, −4.35120059005048981340878569256, −4.17935524508817108632681376472, −3.24714968142473363871996934281, −3.08300653104921256091834257429, −2.22620257266747380171476394618, −2.14409091978434478928599505146, −0.939988324018214861831679105238, −0.61803340199003924739949606625,
0.61803340199003924739949606625, 0.939988324018214861831679105238, 2.14409091978434478928599505146, 2.22620257266747380171476394618, 3.08300653104921256091834257429, 3.24714968142473363871996934281, 4.17935524508817108632681376472, 4.35120059005048981340878569256, 4.56481769788834067678814234164, 5.04542553271678736060633061044, 5.70354752266600993680057293765, 5.83024493637957320724794753856, 6.75436469070911065335904160188, 6.82003096090968680588406798545, 7.38616459036496976541630735904, 7.71679793230319959346671483502, 7.990580720826737929644451502840, 8.315035302192253340514995968085, 9.112554945344679852923605407829, 9.343616970536836988631170542202
