| L(s) = 1 | − 3·3-s + 22.0·5-s + 0.570·7-s + 9·9-s − 14.5·11-s + 13·13-s − 66.2·15-s + 62.8·17-s − 106.·19-s − 1.71·21-s − 175.·23-s + 362.·25-s − 27·27-s − 66.4·29-s − 53.0·31-s + 43.5·33-s + 12.6·35-s − 61.7·37-s − 39·39-s − 244.·41-s + 64.2·43-s + 198.·45-s + 51.2·47-s − 342.·49-s − 188.·51-s − 495.·53-s − 320.·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.97·5-s + 0.0308·7-s + 0.333·9-s − 0.397·11-s + 0.277·13-s − 1.14·15-s + 0.896·17-s − 1.29·19-s − 0.0177·21-s − 1.59·23-s + 2.90·25-s − 0.192·27-s − 0.425·29-s − 0.307·31-s + 0.229·33-s + 0.0608·35-s − 0.274·37-s − 0.160·39-s − 0.932·41-s + 0.227·43-s + 0.658·45-s + 0.159·47-s − 0.999·49-s − 0.517·51-s − 1.28·53-s − 0.786·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 22.0T + 125T^{2} \) |
| 7 | \( 1 - 0.570T + 343T^{2} \) |
| 11 | \( 1 + 14.5T + 1.33e3T^{2} \) |
| 17 | \( 1 - 62.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 106.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 175.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 66.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 53.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + 61.7T + 5.06e4T^{2} \) |
| 41 | \( 1 + 244.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 64.2T + 7.95e4T^{2} \) |
| 47 | \( 1 - 51.2T + 1.03e5T^{2} \) |
| 53 | \( 1 + 495.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 322.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 499.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 145.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.11e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 592.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 124.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 675.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 411.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.63e3T + 9.12e5T^{2} \) |
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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
−8.303860729013096740984128768365, −7.27060942865834336075109373484, −6.26220400051193325216587964258, −5.98167312840418979060079330634, −5.27287136988373463695837340273, −4.40366154969690343931315295770, −3.11089645664373073999737889700, −2.01443822743272049536790746411, −1.47088603652040409415417640908, 0,
1.47088603652040409415417640908, 2.01443822743272049536790746411, 3.11089645664373073999737889700, 4.40366154969690343931315295770, 5.27287136988373463695837340273, 5.98167312840418979060079330634, 6.26220400051193325216587964258, 7.27060942865834336075109373484, 8.303860729013096740984128768365
