| L(s) = 1 | + 2-s + 3-s − 4-s − 2·5-s + 6-s + 4·7-s − 3·8-s + 9-s − 2·10-s − 11-s − 12-s + 2·13-s + 4·14-s − 2·15-s − 16-s + 2·17-s + 18-s + 2·20-s + 4·21-s − 22-s − 8·23-s − 3·24-s − 25-s + 2·26-s + 27-s − 4·28-s + 6·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.894·5-s + 0.408·6-s + 1.51·7-s − 1.06·8-s + 1/3·9-s − 0.632·10-s − 0.301·11-s − 0.288·12-s + 0.554·13-s + 1.06·14-s − 0.516·15-s − 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.447·20-s + 0.872·21-s − 0.213·22-s − 1.66·23-s − 0.612·24-s − 1/5·25-s + 0.392·26-s + 0.192·27-s − 0.755·28-s + 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31713 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31713 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 31 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−15.31768413868313, −14.62100611387198, −14.25452431539289, −13.74650733641698, −13.60205444089597, −12.54467396828532, −12.21777007743641, −11.82623951394482, −11.22407361538637, −10.56199313212551, −10.04402293882880, −9.273134158489500, −8.622761678558068, −8.200644395525594, −7.903054271475804, −7.331614004642958, −6.354038892182273, −5.767498140896010, −5.021964160031570, −4.620997010808923, −3.947561591804752, −3.632877866058986, −2.784747244170823, −1.948583027536287, −1.117699876178469, 0,
1.117699876178469, 1.948583027536287, 2.784747244170823, 3.632877866058986, 3.947561591804752, 4.620997010808923, 5.021964160031570, 5.767498140896010, 6.354038892182273, 7.331614004642958, 7.903054271475804, 8.200644395525594, 8.622761678558068, 9.273134158489500, 10.04402293882880, 10.56199313212551, 11.22407361538637, 11.82623951394482, 12.21777007743641, 12.54467396828532, 13.60205444089597, 13.74650733641698, 14.25452431539289, 14.62100611387198, 15.31768413868313
