| L(s) = 1 | − 2-s + 2.88·3-s + 4-s − 2.88·6-s + 1.88·7-s − 8-s + 5.30·9-s + 6.18·11-s + 2.88·12-s − 1.88·14-s + 16-s − 3·17-s − 5.30·18-s + 4.88·19-s + 5.42·21-s − 6.18·22-s − 0.575·23-s − 2.88·24-s + 6.64·27-s + 1.88·28-s + 5.07·29-s + 7.30·31-s − 32-s + 17.8·33-s + 3·34-s + 5.30·36-s + 9.18·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.66·3-s + 0.5·4-s − 1.17·6-s + 0.711·7-s − 0.353·8-s + 1.76·9-s + 1.86·11-s + 0.831·12-s − 0.502·14-s + 0.250·16-s − 0.727·17-s − 1.25·18-s + 1.12·19-s + 1.18·21-s − 1.31·22-s − 0.120·23-s − 0.588·24-s + 1.27·27-s + 0.355·28-s + 0.941·29-s + 1.31·31-s − 0.176·32-s + 3.10·33-s + 0.514·34-s + 0.884·36-s + 1.51·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.950001642\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.950001642\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 2.88T + 3T^{2} \) |
| 7 | \( 1 - 1.88T + 7T^{2} \) |
| 11 | \( 1 - 6.18T + 11T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 - 4.88T + 19T^{2} \) |
| 23 | \( 1 + 0.575T + 23T^{2} \) |
| 29 | \( 1 - 5.07T + 29T^{2} \) |
| 31 | \( 1 - 7.30T + 31T^{2} \) |
| 37 | \( 1 - 9.18T + 37T^{2} \) |
| 41 | \( 1 - 5.45T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 2.45T + 47T^{2} \) |
| 53 | \( 1 + 11.0T + 53T^{2} \) |
| 59 | \( 1 + 8.83T + 59T^{2} \) |
| 61 | \( 1 - 3.64T + 61T^{2} \) |
| 67 | \( 1 - 3.57T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 11.8T + 73T^{2} \) |
| 79 | \( 1 + 11.1T + 79T^{2} \) |
| 83 | \( 1 - 0.188T + 83T^{2} \) |
| 89 | \( 1 + 11.4T + 89T^{2} \) |
| 97 | \( 1 + 15.7T + 97T^{2} \) |
| 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
−8.121628124551893204897790803720, −7.30631541944612877403236400557, −6.70261721324341612239964172614, −6.01574802273266631100159662934, −4.61386001058717616359999864607, −4.20915087562386301009816096698, −3.22110246684752408933198903503, −2.65827374142801969685897406633, −1.62611692358945464710492520605, −1.15184606277012740608968091174,
1.15184606277012740608968091174, 1.62611692358945464710492520605, 2.65827374142801969685897406633, 3.22110246684752408933198903503, 4.20915087562386301009816096698, 4.61386001058717616359999864607, 6.01574802273266631100159662934, 6.70261721324341612239964172614, 7.30631541944612877403236400557, 8.121628124551893204897790803720
