| L(s) = 1 | + 0.148·2-s − 3-s − 1.97·4-s + 5-s − 0.148·6-s − 0.316·7-s − 0.591·8-s + 9-s + 0.148·10-s + 2.70·11-s + 1.97·12-s − 5.26·13-s − 0.0470·14-s − 15-s + 3.86·16-s + 5.92·17-s + 0.148·18-s − 1.97·20-s + 0.316·21-s + 0.402·22-s − 7.16·23-s + 0.591·24-s + 25-s − 0.782·26-s − 27-s + 0.626·28-s − 2.52·29-s + ⋯ |
| L(s) = 1 | + 0.105·2-s − 0.577·3-s − 0.988·4-s + 0.447·5-s − 0.0606·6-s − 0.119·7-s − 0.209·8-s + 0.333·9-s + 0.0470·10-s + 0.816·11-s + 0.570·12-s − 1.45·13-s − 0.0125·14-s − 0.258·15-s + 0.966·16-s + 1.43·17-s + 0.0350·18-s − 0.442·20-s + 0.0691·21-s + 0.0858·22-s − 1.49·23-s + 0.120·24-s + 0.200·25-s − 0.153·26-s − 0.192·27-s + 0.118·28-s − 0.469·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5415 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 0.148T + 2T^{2} \) |
| 7 | \( 1 + 0.316T + 7T^{2} \) |
| 11 | \( 1 - 2.70T + 11T^{2} \) |
| 13 | \( 1 + 5.26T + 13T^{2} \) |
| 17 | \( 1 - 5.92T + 17T^{2} \) |
| 23 | \( 1 + 7.16T + 23T^{2} \) |
| 29 | \( 1 + 2.52T + 29T^{2} \) |
| 31 | \( 1 + 2.09T + 31T^{2} \) |
| 37 | \( 1 - 2.58T + 37T^{2} \) |
| 41 | \( 1 + 6.94T + 41T^{2} \) |
| 43 | \( 1 + 2.06T + 43T^{2} \) |
| 47 | \( 1 - 7.38T + 47T^{2} \) |
| 53 | \( 1 + 0.121T + 53T^{2} \) |
| 59 | \( 1 - 2.86T + 59T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 67 | \( 1 - 13.2T + 67T^{2} \) |
| 71 | \( 1 - 5.86T + 71T^{2} \) |
| 73 | \( 1 - 13.2T + 73T^{2} \) |
| 79 | \( 1 + 0.652T + 79T^{2} \) |
| 83 | \( 1 + 8.28T + 83T^{2} \) |
| 89 | \( 1 + 10.6T + 89T^{2} \) |
| 97 | \( 1 + 15.1T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.84825213680383955318535408568, −7.05150197416015759248431438618, −6.25511367485531404451869374147, −5.42821257696629816070482580472, −5.12978905972437768192677969871, −4.10684933242906587233762301635, −3.54496572129709934517205226732, −2.29089524776394606854822883316, −1.16866222544377558549149113318, 0,
1.16866222544377558549149113318, 2.29089524776394606854822883316, 3.54496572129709934517205226732, 4.10684933242906587233762301635, 5.12978905972437768192677969871, 5.42821257696629816070482580472, 6.25511367485531404451869374147, 7.05150197416015759248431438618, 7.84825213680383955318535408568
