| L(s) = 1 | + 27·3-s + 333.·5-s + 988.·7-s + 729·9-s − 4.47e3·11-s − 5.59e3·13-s + 8.99e3·15-s + 190.·17-s − 1.23e4·19-s + 2.66e4·21-s + 4.03e4·23-s + 3.29e4·25-s + 1.96e4·27-s + 6.56e4·29-s + 2.26e5·31-s − 1.20e5·33-s + 3.29e5·35-s + 6.08e5·37-s − 1.51e5·39-s + 7.87e4·41-s + 1.05e5·43-s + 2.42e5·45-s + 3.88e5·47-s + 1.53e5·49-s + 5.13e3·51-s − 5.39e4·53-s − 1.49e6·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.19·5-s + 1.08·7-s + 0.333·9-s − 1.01·11-s − 0.706·13-s + 0.688·15-s + 0.00939·17-s − 0.411·19-s + 0.628·21-s + 0.691·23-s + 0.422·25-s + 0.192·27-s + 0.499·29-s + 1.36·31-s − 0.585·33-s + 1.29·35-s + 1.97·37-s − 0.408·39-s + 0.178·41-s + 0.202·43-s + 0.397·45-s + 0.545·47-s + 0.186·49-s + 0.00542·51-s − 0.0497·53-s − 1.20·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(4.114940869\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.114940869\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 27T \) |
| good | 5 | \( 1 - 333.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 988.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 4.47e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 5.59e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 190.T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.23e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 4.03e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 6.56e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.26e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 6.08e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 7.87e4T + 1.94e11T^{2} \) |
| 43 | \( 1 - 1.05e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 3.88e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 5.39e4T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.74e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.42e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.87e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.52e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 4.26e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.27e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 5.13e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.00e7T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.40e7T + 8.07e13T^{2} \) |
| 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
−10.10018471987354758081633950068, −9.293359196441735237591560187725, −8.264270786511294113670582404628, −7.57172361340880716394593846231, −6.31494477275680699810727441089, −5.23176376345608569710032334487, −4.48402068456305320572087087208, −2.72862644772736224795398542743, −2.14149426424356013690816791246, −0.944457678821132456927105406455,
0.944457678821132456927105406455, 2.14149426424356013690816791246, 2.72862644772736224795398542743, 4.48402068456305320572087087208, 5.23176376345608569710032334487, 6.31494477275680699810727441089, 7.57172361340880716394593846231, 8.264270786511294113670582404628, 9.293359196441735237591560187725, 10.10018471987354758081633950068
