| L(s) = 1 | − 2.24·2-s + 0.175·3-s + 3.03·4-s + 1.03·5-s − 0.394·6-s − 1.41·7-s − 2.31·8-s − 2.96·9-s − 2.32·10-s + 0.533·12-s + 4.12·13-s + 3.17·14-s + 0.182·15-s − 0.873·16-s − 1.59·17-s + 6.65·18-s + 2.19·19-s + 3.13·20-s − 0.249·21-s − 5.18·23-s − 0.407·24-s − 3.92·25-s − 9.26·26-s − 1.05·27-s − 4.29·28-s + 2.93·29-s − 0.408·30-s + ⋯ |
| L(s) = 1 | − 1.58·2-s + 0.101·3-s + 1.51·4-s + 0.462·5-s − 0.161·6-s − 0.534·7-s − 0.817·8-s − 0.989·9-s − 0.733·10-s + 0.154·12-s + 1.14·13-s + 0.848·14-s + 0.0470·15-s − 0.218·16-s − 0.385·17-s + 1.56·18-s + 0.503·19-s + 0.701·20-s − 0.0543·21-s − 1.08·23-s − 0.0831·24-s − 0.785·25-s − 1.81·26-s − 0.202·27-s − 0.810·28-s + 0.545·29-s − 0.0745·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6247893017\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6247893017\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 + 2.24T + 2T^{2} \) |
| 3 | \( 1 - 0.175T + 3T^{2} \) |
| 5 | \( 1 - 1.03T + 5T^{2} \) |
| 7 | \( 1 + 1.41T + 7T^{2} \) |
| 13 | \( 1 - 4.12T + 13T^{2} \) |
| 17 | \( 1 + 1.59T + 17T^{2} \) |
| 19 | \( 1 - 2.19T + 19T^{2} \) |
| 23 | \( 1 + 5.18T + 23T^{2} \) |
| 29 | \( 1 - 2.93T + 29T^{2} \) |
| 31 | \( 1 + 9.51T + 31T^{2} \) |
| 37 | \( 1 - 5.19T + 37T^{2} \) |
| 41 | \( 1 - 2.04T + 41T^{2} \) |
| 43 | \( 1 - 3.64T + 43T^{2} \) |
| 47 | \( 1 + 8.09T + 47T^{2} \) |
| 53 | \( 1 - 4.23T + 53T^{2} \) |
| 59 | \( 1 - 11.4T + 59T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 + 12.1T + 71T^{2} \) |
| 73 | \( 1 + 9.82T + 73T^{2} \) |
| 79 | \( 1 + 4.80T + 79T^{2} \) |
| 83 | \( 1 - 3.05T + 83T^{2} \) |
| 89 | \( 1 - 13.8T + 89T^{2} \) |
| 97 | \( 1 - 4.94T + 97T^{2} \) |
| 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
−8.108151709034649270980133349384, −7.46142190834925667680836900037, −6.58993696113374971487506822380, −6.04773723374423450959297255231, −5.45574713669950517687099597632, −4.14641734487595776169303557679, −3.29389486498924385325868907154, −2.36945771824104782434082838840, −1.63612287207827891877003211675, −0.50132558625040679517115003548,
0.50132558625040679517115003548, 1.63612287207827891877003211675, 2.36945771824104782434082838840, 3.29389486498924385325868907154, 4.14641734487595776169303557679, 5.45574713669950517687099597632, 6.04773723374423450959297255231, 6.58993696113374971487506822380, 7.46142190834925667680836900037, 8.108151709034649270980133349384
