| L(s) = 1 | + 3·3-s + 19.4·5-s − 7.48·7-s + 9·9-s − 22.8·11-s − 13·13-s + 58.4·15-s + 67.0·17-s − 16.5·19-s − 22.4·21-s + 175.·23-s + 254.·25-s + 27·27-s + 291.·29-s − 117.·31-s − 68.6·33-s − 145.·35-s − 154.·37-s − 39·39-s − 251.·41-s + 502.·43-s + 175.·45-s + 281.·47-s − 287·49-s + 201.·51-s + 366.·53-s − 446.·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.74·5-s − 0.404·7-s + 0.333·9-s − 0.627·11-s − 0.277·13-s + 1.00·15-s + 0.956·17-s − 0.199·19-s − 0.233·21-s + 1.59·23-s + 2.03·25-s + 0.192·27-s + 1.86·29-s − 0.679·31-s − 0.362·33-s − 0.704·35-s − 0.687·37-s − 0.160·39-s − 0.958·41-s + 1.78·43-s + 0.580·45-s + 0.874·47-s − 0.836·49-s + 0.552·51-s + 0.951·53-s − 1.09·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.423799493\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.423799493\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 13 | \( 1 + 13T \) |
| good | 5 | \( 1 - 19.4T + 125T^{2} \) |
| 7 | \( 1 + 7.48T + 343T^{2} \) |
| 11 | \( 1 + 22.8T + 1.33e3T^{2} \) |
| 17 | \( 1 - 67.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 16.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 175.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 291.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 117.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 154.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 251.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 502.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 281.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 366.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 79.6T + 2.05e5T^{2} \) |
| 61 | \( 1 + 194.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 400.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 528.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 734.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 113.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 933.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.19e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 557.T + 9.12e5T^{2} \) |
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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
−10.27980730776400394067776636620, −9.296797259332597811529216426574, −8.764138241710498833210539088719, −7.49909351511723033663309175861, −6.58907235768349528196182927778, −5.64459754672947573197855165095, −4.83736363374318329539729612758, −3.15153143650535655539178554888, −2.40507717432568676248969787613, −1.15223979646295715452197509725,
1.15223979646295715452197509725, 2.40507717432568676248969787613, 3.15153143650535655539178554888, 4.83736363374318329539729612758, 5.64459754672947573197855165095, 6.58907235768349528196182927778, 7.49909351511723033663309175861, 8.764138241710498833210539088719, 9.296797259332597811529216426574, 10.27980730776400394067776636620
