| L(s) = 1 | − 27·3-s + 248.·5-s + 729·9-s − 2.96e3·11-s − 1.17e4·13-s − 6.70e3·15-s − 2.26e4·17-s − 2.27e4·19-s + 4.62e4·23-s − 1.64e4·25-s − 1.96e4·27-s − 1.14e5·29-s − 1.41e5·31-s + 8.01e4·33-s + 3.86e5·37-s + 3.16e5·39-s − 1.79e5·41-s + 4.75e5·43-s + 1.81e5·45-s − 2.17e5·47-s + 6.12e5·51-s + 1.98e6·53-s − 7.37e5·55-s + 6.14e5·57-s − 6.21e5·59-s + 2.82e6·61-s − 2.91e6·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.888·5-s + 0.333·9-s − 0.672·11-s − 1.48·13-s − 0.513·15-s − 1.11·17-s − 0.761·19-s + 0.793·23-s − 0.210·25-s − 0.192·27-s − 0.874·29-s − 0.852·31-s + 0.388·33-s + 1.25·37-s + 0.854·39-s − 0.407·41-s + 0.912·43-s + 0.296·45-s − 0.305·47-s + 0.646·51-s + 1.83·53-s − 0.597·55-s + 0.439·57-s − 0.393·59-s + 1.59·61-s − 1.31·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.140588534\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.140588534\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 27T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 248.T + 7.81e4T^{2} \) |
| 11 | \( 1 + 2.96e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.17e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.26e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.27e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 4.62e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.14e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.41e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.86e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 1.79e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 4.75e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 2.17e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.98e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 6.21e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.82e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.16e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 4.03e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.64e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 2.13e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 5.90e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 5.17e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 3.70e6T + 8.07e13T^{2} \) |
| 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
−9.650569158209455966238569234194, −8.900509167933260419612258596727, −7.60859981894863558786993166423, −6.85714101573619682500541004939, −5.84710180289093952537229717259, −5.13137851982693831117795895219, −4.21669866148626745770699385814, −2.58841684782817055432475662442, −1.93562092494232212087260294993, −0.44406992702826232265840004663,
0.44406992702826232265840004663, 1.93562092494232212087260294993, 2.58841684782817055432475662442, 4.21669866148626745770699385814, 5.13137851982693831117795895219, 5.84710180289093952537229717259, 6.85714101573619682500541004939, 7.60859981894863558786993166423, 8.900509167933260419612258596727, 9.650569158209455966238569234194
