| L(s) = 1 | + 2·2-s + 4.88·3-s + 4·4-s + 9.76·6-s + 32.3·7-s + 8·8-s − 3.16·9-s + 47.0·11-s + 19.5·12-s − 9.97·13-s + 64.7·14-s + 16·16-s + 45.7·17-s − 6.32·18-s − 117.·19-s + 158.·21-s + 94.0·22-s + 8.72·23-s + 39.0·24-s − 19.9·26-s − 147.·27-s + 129.·28-s + 233.·29-s + 262.·31-s + 32·32-s + 229.·33-s + 91.4·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.939·3-s + 0.5·4-s + 0.664·6-s + 1.74·7-s + 0.353·8-s − 0.117·9-s + 1.28·11-s + 0.469·12-s − 0.212·13-s + 1.23·14-s + 0.250·16-s + 0.652·17-s − 0.0828·18-s − 1.41·19-s + 1.64·21-s + 0.911·22-s + 0.0791·23-s + 0.332·24-s − 0.150·26-s − 1.04·27-s + 0.874·28-s + 1.49·29-s + 1.51·31-s + 0.176·32-s + 1.21·33-s + 0.461·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1250 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.396842378\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.396842378\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 4.88T + 27T^{2} \) |
| 7 | \( 1 - 32.3T + 343T^{2} \) |
| 11 | \( 1 - 47.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 9.97T + 2.19e3T^{2} \) |
| 17 | \( 1 - 45.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 117.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 8.72T + 1.21e4T^{2} \) |
| 29 | \( 1 - 233.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 262.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 14.8T + 5.06e4T^{2} \) |
| 41 | \( 1 + 256.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 81.5T + 7.95e4T^{2} \) |
| 47 | \( 1 - 218.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 324.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 524.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 251.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 740.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 379.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 367.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 916.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 439.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 381.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 672.T + 9.12e5T^{2} \) |
| 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
−9.069519774188237374610937061881, −8.352357856116072435148712471155, −7.931767415757626656400764917203, −6.81959529790739111676295976511, −5.94392045389505596545074494242, −4.75865059907843518893401580026, −4.26800060990116812308874694470, −3.14184242365814160834130211639, −2.13141591359017550618981013051, −1.25667329965269707724944843048,
1.25667329965269707724944843048, 2.13141591359017550618981013051, 3.14184242365814160834130211639, 4.26800060990116812308874694470, 4.75865059907843518893401580026, 5.94392045389505596545074494242, 6.81959529790739111676295976511, 7.931767415757626656400764917203, 8.352357856116072435148712471155, 9.069519774188237374610937061881
