L(s) = 1 | − 3-s − 2·5-s − 7-s + 9-s − 4·11-s + 2·13-s + 2·15-s − 6·17-s + 19-s + 21-s − 25-s − 27-s − 10·29-s + 8·31-s + 4·33-s + 2·35-s − 2·37-s − 2·39-s + 6·41-s + 4·43-s − 2·45-s − 8·47-s + 49-s + 6·51-s + 6·53-s + 8·55-s − 57-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s − 1.20·11-s + 0.554·13-s + 0.516·15-s − 1.45·17-s + 0.229·19-s + 0.218·21-s − 1/5·25-s − 0.192·27-s − 1.85·29-s + 1.43·31-s + 0.696·33-s + 0.338·35-s − 0.328·37-s − 0.320·39-s + 0.937·41-s + 0.609·43-s − 0.298·45-s − 1.16·47-s + 1/7·49-s + 0.840·51-s + 0.824·53-s + 1.07·55-s − 0.132·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6175010944\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6175010944\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 - T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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
−8.592755631704368409772717004015, −7.80947253152615684676372631812, −7.26744003625055323702271169821, −6.36508488266078460320337894920, −5.69078191275565371724786967114, −4.74927395966529356148729750817, −4.07607860823459720967336033672, −3.15139337097511963871769455963, −2.06038368310027988339518517086, −0.46429122192493449796476089591,
0.46429122192493449796476089591, 2.06038368310027988339518517086, 3.15139337097511963871769455963, 4.07607860823459720967336033672, 4.74927395966529356148729750817, 5.69078191275565371724786967114, 6.36508488266078460320337894920, 7.26744003625055323702271169821, 7.80947253152615684676372631812, 8.592755631704368409772717004015