| L(s) = 1 | + 2-s − 0.0938·3-s + 4-s − 0.0938·6-s − 7-s + 8-s − 2.99·9-s + 2.95·11-s − 0.0938·12-s − 3.70·13-s − 14-s + 16-s − 17-s − 2.99·18-s + 3.33·19-s + 0.0938·21-s + 2.95·22-s − 4.02·23-s − 0.0938·24-s − 3.70·26-s + 0.562·27-s − 28-s − 1.83·29-s + 8.40·31-s + 32-s − 0.277·33-s − 34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.0541·3-s + 0.5·4-s − 0.0382·6-s − 0.377·7-s + 0.353·8-s − 0.997·9-s + 0.892·11-s − 0.0270·12-s − 1.02·13-s − 0.267·14-s + 0.250·16-s − 0.242·17-s − 0.705·18-s + 0.764·19-s + 0.0204·21-s + 0.630·22-s − 0.840·23-s − 0.0191·24-s − 0.727·26-s + 0.108·27-s − 0.188·28-s − 0.340·29-s + 1.50·31-s + 0.176·32-s − 0.0483·33-s − 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + 0.0938T + 3T^{2} \) |
| 11 | \( 1 - 2.95T + 11T^{2} \) |
| 13 | \( 1 + 3.70T + 13T^{2} \) |
| 19 | \( 1 - 3.33T + 19T^{2} \) |
| 23 | \( 1 + 4.02T + 23T^{2} \) |
| 29 | \( 1 + 1.83T + 29T^{2} \) |
| 31 | \( 1 - 8.40T + 31T^{2} \) |
| 37 | \( 1 - 9.60T + 37T^{2} \) |
| 41 | \( 1 + 6.25T + 41T^{2} \) |
| 43 | \( 1 + 8.97T + 43T^{2} \) |
| 47 | \( 1 + 6.60T + 47T^{2} \) |
| 53 | \( 1 + 13.1T + 53T^{2} \) |
| 59 | \( 1 - 4.82T + 59T^{2} \) |
| 61 | \( 1 + 14.3T + 61T^{2} \) |
| 67 | \( 1 - 7.41T + 67T^{2} \) |
| 71 | \( 1 + 6.15T + 71T^{2} \) |
| 73 | \( 1 - 5.53T + 73T^{2} \) |
| 79 | \( 1 + 3.94T + 79T^{2} \) |
| 83 | \( 1 + 0.858T + 83T^{2} \) |
| 89 | \( 1 + 2.26T + 89T^{2} \) |
| 97 | \( 1 - 7.50T + 97T^{2} \) |
| 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
−7.80355637577230442800568642035, −6.62183371069545249727361514487, −6.45858968072603371861724503542, −5.52350641763212375286128426178, −4.86234875051398613243332654458, −4.09333174962005913132649625943, −3.17234045633390977946909414879, −2.62300646806034423758623031127, −1.48368473997653219576751488898, 0,
1.48368473997653219576751488898, 2.62300646806034423758623031127, 3.17234045633390977946909414879, 4.09333174962005913132649625943, 4.86234875051398613243332654458, 5.52350641763212375286128426178, 6.45858968072603371861724503542, 6.62183371069545249727361514487, 7.80355637577230442800568642035
