| L(s) = 1 | − 8·2-s + 8.12·3-s + 64·4-s − 64.9·6-s + 343·7-s − 512·8-s − 2.12e3·9-s − 1.56e3·11-s + 519.·12-s − 6.85e3·13-s − 2.74e3·14-s + 4.09e3·16-s − 3.03e4·17-s + 1.69e4·18-s − 7.15e3·19-s + 2.78e3·21-s + 1.25e4·22-s − 1.21e4·23-s − 4.15e3·24-s + 5.48e4·26-s − 3.49e4·27-s + 2.19e4·28-s + 7.29e3·29-s − 4.40e3·31-s − 3.27e4·32-s − 1.27e4·33-s + 2.42e5·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.173·3-s + 0.5·4-s − 0.122·6-s + 0.377·7-s − 0.353·8-s − 0.969·9-s − 0.355·11-s + 0.0868·12-s − 0.865·13-s − 0.267·14-s + 0.250·16-s − 1.49·17-s + 0.685·18-s − 0.239·19-s + 0.0656·21-s + 0.251·22-s − 0.207·23-s − 0.0614·24-s + 0.611·26-s − 0.342·27-s + 0.188·28-s + 0.0555·29-s − 0.0265·31-s − 0.176·32-s − 0.0617·33-s + 1.05·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.8590811416\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8590811416\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 8T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 343T \) |
| good | 3 | \( 1 - 8.12T + 2.18e3T^{2} \) |
| 11 | \( 1 + 1.56e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 6.85e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 3.03e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 7.15e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 1.21e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 7.29e3T + 1.72e10T^{2} \) |
| 31 | \( 1 + 4.40e3T + 2.75e10T^{2} \) |
| 37 | \( 1 - 4.77e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 4.62e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 1.77e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 4.24e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 6.85e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 3.43e3T + 2.48e12T^{2} \) |
| 61 | \( 1 + 3.02e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 4.02e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 2.50e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 5.59e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 1.90e5T + 1.92e13T^{2} \) |
| 83 | \( 1 - 4.96e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 8.40e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 3.36e6T + 8.07e13T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.25417926303824477968864863318, −9.220248991125228074855001005958, −8.498904889899730010877027307510, −7.66757877176266893248954833091, −6.63382868890929893398712473361, −5.53615763620451818326796654729, −4.36445880410421186338023396668, −2.80168371736694611245281211964, −2.04079360289676906013358162437, −0.45763350937143896426776080007,
0.45763350937143896426776080007, 2.04079360289676906013358162437, 2.80168371736694611245281211964, 4.36445880410421186338023396668, 5.53615763620451818326796654729, 6.63382868890929893398712473361, 7.66757877176266893248954833091, 8.498904889899730010877027307510, 9.220248991125228074855001005958, 10.25417926303824477968864863318
