| L(s) = 1 | − 3·3-s − 4.87·5-s + 25.6·7-s + 9·9-s − 59.7·11-s + 13·13-s + 14.6·15-s − 75.4·17-s + 116.·19-s − 76.8·21-s + 90.5·23-s − 101.·25-s − 27·27-s − 187.·29-s + 225.·31-s + 179.·33-s − 124.·35-s + 290.·37-s − 39·39-s − 191.·41-s − 326.·43-s − 43.8·45-s + 406.·47-s + 313.·49-s + 226.·51-s − 426.·53-s + 291.·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.436·5-s + 1.38·7-s + 0.333·9-s − 1.63·11-s + 0.277·13-s + 0.251·15-s − 1.07·17-s + 1.40·19-s − 0.798·21-s + 0.820·23-s − 0.809·25-s − 0.192·27-s − 1.19·29-s + 1.30·31-s + 0.945·33-s − 0.603·35-s + 1.29·37-s − 0.160·39-s − 0.728·41-s − 1.15·43-s − 0.145·45-s + 1.26·47-s + 0.912·49-s + 0.621·51-s − 1.10·53-s + 0.714·55-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
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 4.87T + 125T^{2} \) |
| 7 | \( 1 - 25.6T + 343T^{2} \) |
| 11 | \( 1 + 59.7T + 1.33e3T^{2} \) |
| 17 | \( 1 + 75.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 116.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 90.5T + 1.21e4T^{2} \) |
| 29 | \( 1 + 187.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 225.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 290.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 191.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 326.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 406.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 426.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 331.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 524.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 968.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 8.39T + 3.57e5T^{2} \) |
| 73 | \( 1 + 903.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.15e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 952.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.05e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 90.7T + 9.12e5T^{2} \) |
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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
−9.958573149084761134830499635690, −8.765543794156131513404775678444, −7.80402029904328600392745138027, −7.40191900613946169933490094297, −5.94842591727566714114340430432, −5.06108999293697415676033783605, −4.42670405543754687175605458625, −2.86884636106324375903266040912, −1.48292116764370884038941368412, 0,
1.48292116764370884038941368412, 2.86884636106324375903266040912, 4.42670405543754687175605458625, 5.06108999293697415676033783605, 5.94842591727566714114340430432, 7.40191900613946169933490094297, 7.80402029904328600392745138027, 8.765543794156131513404775678444, 9.958573149084761134830499635690
