| L(s) = 1 | + 2-s − 3-s + 4-s + 3.03·5-s − 6-s + 8-s + 9-s + 3.03·10-s − 1.03·11-s − 12-s + 3.22·13-s − 3.03·15-s + 16-s + 3.22·17-s + 18-s − 2.85·19-s + 3.03·20-s − 1.03·22-s + 7.89·23-s − 24-s + 4.22·25-s + 3.22·26-s − 27-s + 9.14·29-s − 3.03·30-s + 31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.35·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s + 0.960·10-s − 0.312·11-s − 0.288·12-s + 0.895·13-s − 0.784·15-s + 0.250·16-s + 0.782·17-s + 0.235·18-s − 0.655·19-s + 0.679·20-s − 0.221·22-s + 1.64·23-s − 0.204·24-s + 0.845·25-s + 0.632·26-s − 0.192·27-s + 1.69·29-s − 0.554·30-s + 0.179·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.216462729\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.216462729\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 5 | \( 1 - 3.03T + 5T^{2} \) |
| 11 | \( 1 + 1.03T + 11T^{2} \) |
| 13 | \( 1 - 3.22T + 13T^{2} \) |
| 17 | \( 1 - 3.22T + 17T^{2} \) |
| 19 | \( 1 + 2.85T + 19T^{2} \) |
| 23 | \( 1 - 7.89T + 23T^{2} \) |
| 29 | \( 1 - 9.14T + 29T^{2} \) |
| 37 | \( 1 + 4.58T + 37T^{2} \) |
| 41 | \( 1 + 0.179T + 41T^{2} \) |
| 43 | \( 1 + 11.0T + 43T^{2} \) |
| 47 | \( 1 - 9.62T + 47T^{2} \) |
| 53 | \( 1 - 8.76T + 53T^{2} \) |
| 59 | \( 1 - 0.689T + 59T^{2} \) |
| 61 | \( 1 + 11.0T + 61T^{2} \) |
| 67 | \( 1 - 4.84T + 67T^{2} \) |
| 71 | \( 1 + 15.1T + 71T^{2} \) |
| 73 | \( 1 + 11.9T + 73T^{2} \) |
| 79 | \( 1 + 0.347T + 79T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 - 14.1T + 89T^{2} \) |
| 97 | \( 1 + 13.8T + 97T^{2} \) |
| 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
−7.41717596048144886980188511353, −6.80433312140486788630855361283, −6.13583279030084307723744963427, −5.74495161906496661239703304808, −5.03848724328490391785632156047, −4.48937774391407939088105049321, −3.37618185670401382414100035708, −2.71927037957713419563283978337, −1.72738968550599773716908830720, −0.984679059442436294209931885023,
0.984679059442436294209931885023, 1.72738968550599773716908830720, 2.71927037957713419563283978337, 3.37618185670401382414100035708, 4.48937774391407939088105049321, 5.03848724328490391785632156047, 5.74495161906496661239703304808, 6.13583279030084307723744963427, 6.80433312140486788630855361283, 7.41717596048144886980188511353
