| L(s) = 1 | − 4.40·2-s + 3·3-s + 11.4·4-s − 5·5-s − 13.2·6-s − 17.2·7-s − 15.0·8-s + 9·9-s + 22.0·10-s + 34.2·12-s − 25.9·13-s + 76.0·14-s − 15·15-s − 24.9·16-s − 45.4·17-s − 39.6·18-s − 17.3·19-s − 57.0·20-s − 51.8·21-s − 54.3·23-s − 45.2·24-s + 25·25-s + 114.·26-s + 27·27-s − 197.·28-s − 201.·29-s + 66.1·30-s + ⋯ |
| L(s) = 1 | − 1.55·2-s + 0.577·3-s + 1.42·4-s − 0.447·5-s − 0.899·6-s − 0.932·7-s − 0.665·8-s + 0.333·9-s + 0.696·10-s + 0.824·12-s − 0.553·13-s + 1.45·14-s − 0.258·15-s − 0.389·16-s − 0.648·17-s − 0.519·18-s − 0.209·19-s − 0.638·20-s − 0.538·21-s − 0.492·23-s − 0.384·24-s + 0.200·25-s + 0.862·26-s + 0.192·27-s − 1.33·28-s − 1.29·29-s + 0.402·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3862754509\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3862754509\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 5 | \( 1 + 5T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 4.40T + 8T^{2} \) |
| 7 | \( 1 + 17.2T + 343T^{2} \) |
| 13 | \( 1 + 25.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 45.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 17.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 54.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 201.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 71.2T + 2.97e4T^{2} \) |
| 37 | \( 1 - 30.1T + 5.06e4T^{2} \) |
| 41 | \( 1 + 339.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 142.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 387.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 207.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 683.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 173.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 35.1T + 3.00e5T^{2} \) |
| 71 | \( 1 - 331.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.04e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 302.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 831.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 18.4T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.17e3T + 9.12e5T^{2} \) |
| 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
−8.951376852332243773741513838607, −8.278444813262194026353405601669, −7.50076439721451892583996687386, −6.95423230887235451738474782927, −6.10739521064706489395102133613, −4.68277580068362182954570712620, −3.65175197072564017567601588874, −2.63726958673393140239236292508, −1.71512233230706763753196427151, −0.34928161401675958095280351760,
0.34928161401675958095280351760, 1.71512233230706763753196427151, 2.63726958673393140239236292508, 3.65175197072564017567601588874, 4.68277580068362182954570712620, 6.10739521064706489395102133613, 6.95423230887235451738474782927, 7.50076439721451892583996687386, 8.278444813262194026353405601669, 8.951376852332243773741513838607
