| L(s) = 1 | + 256·4-s − 1.15e3·5-s − 1.46e4·11-s + 6.55e4·16-s − 2.94e5·20-s + 5.31e5·23-s + 9.34e5·25-s − 1.54e6·31-s + 7.16e5·37-s − 3.74e6·44-s + 6.08e6·47-s + 5.76e6·49-s + 1.52e7·53-s + 1.68e7·55-s + 4.10e6·59-s + 1.67e7·64-s + 1.98e7·67-s − 7.04e6·71-s − 7.54e7·80-s + 8.41e7·89-s + 1.36e8·92-s − 8.11e7·97-s + 2.39e8·100-s − 3.62e6·103-s − 1.01e8·113-s − 6.12e8·115-s + ⋯ |
| L(s) = 1 | + 4-s − 1.84·5-s − 11-s + 16-s − 1.84·20-s + 1.90·23-s + 2.39·25-s − 1.66·31-s + 0.382·37-s − 44-s + 1.24·47-s + 49-s + 1.93·53-s + 1.84·55-s + 0.338·59-s + 64-s + 0.982·67-s − 0.277·71-s − 1.84·80-s + 1.34·89-s + 1.90·92-s − 0.916·97-s + 2.39·100-s − 0.0322·103-s − 0.624·113-s − 3.49·115-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.591340511\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.591340511\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + p^{4} T \) |
| good | 2 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 5 | \( 1 + 1151 T + p^{8} T^{2} \) |
| 7 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 13 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 17 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 19 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 23 | \( 1 - 531793 T + p^{8} T^{2} \) |
| 29 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 31 | \( 1 + 1541233 T + p^{8} T^{2} \) |
| 37 | \( 1 - 716447 T + p^{8} T^{2} \) |
| 41 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 43 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 47 | \( 1 - 6080638 T + p^{8} T^{2} \) |
| 53 | \( 1 - 15265438 T + p^{8} T^{2} \) |
| 59 | \( 1 - 4101553 T + p^{8} T^{2} \) |
| 61 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 67 | \( 1 - 19806767 T + p^{8} T^{2} \) |
| 71 | \( 1 + 7043087 T + p^{8} T^{2} \) |
| 73 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 79 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 83 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 89 | \( 1 - 84100993 T + p^{8} T^{2} \) |
| 97 | \( 1 + 81155713 T + p^{8} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.14137087328385802258826632174, −11.19439085101696741204618287085, −10.62943513321102072921375512272, −8.738627256085313969954019262152, −7.58830407518295512995371641264, −7.07113928930912776049405793733, −5.29302210572638443674360194120, −3.78524835754885102650873961033, −2.67163070383545103821817819942, −0.72650840871199448936161405043,
0.72650840871199448936161405043, 2.67163070383545103821817819942, 3.78524835754885102650873961033, 5.29302210572638443674360194120, 7.07113928930912776049405793733, 7.58830407518295512995371641264, 8.738627256085313969954019262152, 10.62943513321102072921375512272, 11.19439085101696741204618287085, 12.14137087328385802258826632174
