| L(s) = 1 | − 5·5-s + 22·7-s − 9·11-s − 17·13-s + 150·17-s − 8·19-s + 183·23-s + 129·29-s + 187·31-s − 110·35-s − 68·37-s + 264·41-s − 443·43-s + 609·47-s + 343·49-s + 456·53-s + 45·55-s + 60·59-s + 454·61-s + 85·65-s + 244·67-s − 888·71-s + 796·73-s − 198·77-s + 349·79-s + 1.03e3·83-s − 750·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.18·7-s − 0.246·11-s − 0.362·13-s + 2.14·17-s − 0.0965·19-s + 1.65·23-s + 0.826·29-s + 1.08·31-s − 0.531·35-s − 0.302·37-s + 1.00·41-s − 1.57·43-s + 1.89·47-s + 49-s + 1.18·53-s + 0.110·55-s + 0.132·59-s + 0.952·61-s + 0.162·65-s + 0.444·67-s − 1.48·71-s + 1.27·73-s − 0.293·77-s + 0.497·79-s + 1.37·83-s − 0.957·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.555948692\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.555948692\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 22 T + 141 T^{2} - 22 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 9 T - 1250 T^{2} + 9 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 17 T - 1908 T^{2} + 17 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 75 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 183 T + 21322 T^{2} - 183 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 129 T - 7748 T^{2} - 129 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 187 T + 5178 T^{2} - 187 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 34 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 264 T + 775 T^{2} - 264 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 443 T + 116742 T^{2} + 443 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 609 T + 267058 T^{2} - 609 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 228 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 60 T - 201779 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 454 T - 20865 T^{2} - 454 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 244 T - 241227 T^{2} - 244 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 444 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 398 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 349 T - 371238 T^{2} - 349 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 1038 T + 505657 T^{2} - 1038 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 852 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 914 T - 77277 T^{2} + 914 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.927160289944172325203777224848, −8.887021347989308177739519677577, −8.434849524621675052822974007102, −7.935457598451804144186668615434, −7.56378067924706852336035093852, −7.56352253279900984347434398898, −6.73405443485970310721049845661, −6.66622860099694755241341941637, −5.78772693843080610431410256309, −5.46817162028381580440138851295, −5.09518968601489752763870683793, −4.83151475442300871000364450268, −4.10719458588245643611938530978, −3.91717419039542342930871540096, −3.00673134517374779776956232836, −2.93671301088842772093354486414, −2.20335936984373662627277247895, −1.51156191971170039694481270940, −0.824035797658204898708101768928, −0.73221605902118199363277345128,
0.73221605902118199363277345128, 0.824035797658204898708101768928, 1.51156191971170039694481270940, 2.20335936984373662627277247895, 2.93671301088842772093354486414, 3.00673134517374779776956232836, 3.91717419039542342930871540096, 4.10719458588245643611938530978, 4.83151475442300871000364450268, 5.09518968601489752763870683793, 5.46817162028381580440138851295, 5.78772693843080610431410256309, 6.66622860099694755241341941637, 6.73405443485970310721049845661, 7.56352253279900984347434398898, 7.56378067924706852336035093852, 7.935457598451804144186668615434, 8.434849524621675052822974007102, 8.887021347989308177739519677577, 8.927160289944172325203777224848
