| L(s) = 1 | − 6·3-s − 6·7-s + 27·9-s + 42·11-s + 78·13-s + 102·17-s − 56·19-s + 36·21-s + 48·23-s − 108·27-s − 318·29-s − 52·31-s − 252·33-s + 306·37-s − 468·39-s − 408·41-s − 120·43-s − 180·47-s − 290·49-s − 612·51-s + 402·53-s + 336·57-s + 186·59-s + 340·61-s − 162·63-s − 732·67-s − 288·69-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.323·7-s + 9-s + 1.15·11-s + 1.66·13-s + 1.45·17-s − 0.676·19-s + 0.374·21-s + 0.435·23-s − 0.769·27-s − 2.03·29-s − 0.301·31-s − 1.32·33-s + 1.35·37-s − 1.92·39-s − 1.55·41-s − 0.425·43-s − 0.558·47-s − 0.845·49-s − 1.68·51-s + 1.04·53-s + 0.780·57-s + 0.410·59-s + 0.713·61-s − 0.323·63-s − 1.33·67-s − 0.502·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.602999632\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.602999632\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $D_{4}$ | \( 1 + 6 T + 326 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 42 T + 2734 T^{2} - 42 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 6 p T + 5546 T^{2} - 6 p^{4} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 6 p T + 11402 T^{2} - 6 p^{4} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 56 T + 8598 T^{2} + 56 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 48 T + 24254 T^{2} - 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 318 T + 55978 T^{2} + 318 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 52 T + 58782 T^{2} + 52 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 306 T + 94826 T^{2} - 306 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 408 T + 177982 T^{2} + 408 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 120 T + 68150 T^{2} + 120 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 180 T + 128990 T^{2} + 180 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 402 T + 269234 T^{2} - 402 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 186 T + 419038 T^{2} - 186 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 340 T + 388398 T^{2} - 340 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 732 T + 698582 T^{2} + 732 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 36 T + 384046 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 1332 T + 1102034 T^{2} - 1332 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 380 T + 99678 T^{2} + 380 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 984 T + 942182 T^{2} - 984 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 1116 T + 1508758 T^{2} - 1116 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 768 T + 1382402 T^{2} + 768 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.478131657047385873086771160621, −9.365006049421048729686998040366, −8.724237432082986856781432388814, −8.489527145532970048176229211555, −7.73911056024936449867241792804, −7.62795954069138429535201246327, −6.84372110389886155498715598437, −6.64004705073133077799422442017, −6.15131461203116827960301539820, −5.95799084770097863095443770661, −5.29361751918668072950608277121, −5.17382705644969217853794916366, −4.25407457691096052783135073048, −4.01241031346673020480048410281, −3.35443132040750663178276035261, −3.28569540606277144488958352111, −1.91641532286767805795984862248, −1.69619500963758275343443964988, −0.892085465030633368401629993636, −0.54384344756174017117190352172,
0.54384344756174017117190352172, 0.892085465030633368401629993636, 1.69619500963758275343443964988, 1.91641532286767805795984862248, 3.28569540606277144488958352111, 3.35443132040750663178276035261, 4.01241031346673020480048410281, 4.25407457691096052783135073048, 5.17382705644969217853794916366, 5.29361751918668072950608277121, 5.95799084770097863095443770661, 6.15131461203116827960301539820, 6.64004705073133077799422442017, 6.84372110389886155498715598437, 7.62795954069138429535201246327, 7.73911056024936449867241792804, 8.489527145532970048176229211555, 8.724237432082986856781432388814, 9.365006049421048729686998040366, 9.478131657047385873086771160621
