| L(s) = 1 | + 4.04·2-s + 8.39·4-s − 0.434·5-s + 6.63·7-s + 1.59·8-s − 1.76·10-s − 27.7·11-s + 56.0·13-s + 26.8·14-s − 60.6·16-s + 37.6·17-s + 31.7·19-s − 3.65·20-s − 112.·22-s + 192.·23-s − 124.·25-s + 226.·26-s + 55.7·28-s + 71.5·29-s + 98.8·31-s − 258.·32-s + 152.·34-s − 2.88·35-s + 181.·37-s + 128.·38-s − 0.695·40-s + 353.·41-s + ⋯ |
| L(s) = 1 | + 1.43·2-s + 1.04·4-s − 0.0389·5-s + 0.358·7-s + 0.0706·8-s − 0.0556·10-s − 0.761·11-s + 1.19·13-s + 0.513·14-s − 0.948·16-s + 0.537·17-s + 0.382·19-s − 0.0408·20-s − 1.08·22-s + 1.74·23-s − 0.998·25-s + 1.71·26-s + 0.376·28-s + 0.458·29-s + 0.572·31-s − 1.42·32-s + 0.769·34-s − 0.0139·35-s + 0.806·37-s + 0.547·38-s − 0.00274·40-s + 1.34·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1161 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.020608136\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.020608136\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 43 | \( 1 - 43T \) |
| good | 2 | \( 1 - 4.04T + 8T^{2} \) |
| 5 | \( 1 + 0.434T + 125T^{2} \) |
| 7 | \( 1 - 6.63T + 343T^{2} \) |
| 11 | \( 1 + 27.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 56.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 37.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 31.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 192.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 71.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 98.8T + 2.97e4T^{2} \) |
| 37 | \( 1 - 181.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 353.T + 6.89e4T^{2} \) |
| 47 | \( 1 + 224.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 726.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 646.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 159.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 115.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 938.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 37.8T + 3.89e5T^{2} \) |
| 79 | \( 1 - 468.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.32e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 133.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.08e3T + 9.12e5T^{2} \) |
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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
−9.410631144214691470190738251379, −8.486949764106299822913836268634, −7.62626470240862185191626093059, −6.62593695934687490288070468952, −5.76001148607398310321736676329, −5.14233090515986192506348258585, −4.23253013904961418340349047454, −3.34781683744656257186978400246, −2.47754382241101066616755419842, −0.968774610940925758272600356729,
0.968774610940925758272600356729, 2.47754382241101066616755419842, 3.34781683744656257186978400246, 4.23253013904961418340349047454, 5.14233090515986192506348258585, 5.76001148607398310321736676329, 6.62593695934687490288070468952, 7.62626470240862185191626093059, 8.486949764106299822913836268634, 9.410631144214691470190738251379
