| L(s) = 1 | − 3·3-s + 2.70·5-s + 9·9-s − 64.1·11-s − 70.8·13-s − 8.11·15-s + 14.7·17-s + 159.·19-s + 172.·23-s − 117.·25-s − 27·27-s + 18.2·29-s + 149.·31-s + 192.·33-s + 41.7·37-s + 212.·39-s + 50.2·41-s + 388·43-s + 24.3·45-s − 494.·47-s − 44.1·51-s + 469.·53-s − 173.·55-s − 477.·57-s + 343.·59-s + 72.4·61-s − 191.·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.241·5-s + 0.333·9-s − 1.75·11-s − 1.51·13-s − 0.139·15-s + 0.209·17-s + 1.92·19-s + 1.56·23-s − 0.941·25-s − 0.192·27-s + 0.116·29-s + 0.865·31-s + 1.01·33-s + 0.185·37-s + 0.872·39-s + 0.191·41-s + 1.37·43-s + 0.0806·45-s − 1.53·47-s − 0.121·51-s + 1.21·53-s − 0.425·55-s − 1.10·57-s + 0.757·59-s + 0.152·61-s − 0.365·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 3T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 2.70T + 125T^{2} \) |
| 11 | \( 1 + 64.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 70.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 14.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 159.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 172.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 18.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 149.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 41.7T + 5.06e4T^{2} \) |
| 41 | \( 1 - 50.2T + 6.89e4T^{2} \) |
| 43 | \( 1 - 388T + 7.95e4T^{2} \) |
| 47 | \( 1 + 494.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 469.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 343.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 72.4T + 2.26e5T^{2} \) |
| 67 | \( 1 + 293.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 629.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 696.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 640.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.07e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 667.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.30e3T + 9.12e5T^{2} \) |
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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
−7.953981214812701188168046414138, −7.52123467683956063830001157016, −6.82301284197773170078173778029, −5.50259619814490674494861324957, −5.33285926104352721304926175349, −4.49855160803122995793156022666, −3.05339237304820829365964583369, −2.46590024774630112825318802042, −1.05244787555675405115651522658, 0,
1.05244787555675405115651522658, 2.46590024774630112825318802042, 3.05339237304820829365964583369, 4.49855160803122995793156022666, 5.33285926104352721304926175349, 5.50259619814490674494861324957, 6.82301284197773170078173778029, 7.52123467683956063830001157016, 7.953981214812701188168046414138
