| L(s) = 1 | + 7.17·2-s − 76.4·4-s + 94.6·5-s − 1.41e3·7-s − 1.46e3·8-s + 679.·10-s + 467.·11-s − 9.66e3·13-s − 1.01e4·14-s − 751.·16-s + 1.42e3·17-s − 8.25e3·19-s − 7.23e3·20-s + 3.35e3·22-s − 5.57e4·23-s − 6.91e4·25-s − 6.93e4·26-s + 1.08e5·28-s − 1.51e5·29-s − 4.38e4·31-s + 1.82e5·32-s + 1.02e4·34-s − 1.34e5·35-s − 2.99e5·37-s − 5.92e4·38-s − 1.38e5·40-s − 3.06e5·41-s + ⋯ |
| L(s) = 1 | + 0.634·2-s − 0.597·4-s + 0.338·5-s − 1.56·7-s − 1.01·8-s + 0.214·10-s + 0.105·11-s − 1.22·13-s − 0.992·14-s − 0.0458·16-s + 0.0704·17-s − 0.275·19-s − 0.202·20-s + 0.0672·22-s − 0.955·23-s − 0.885·25-s − 0.774·26-s + 0.934·28-s − 1.15·29-s − 0.264·31-s + 0.984·32-s + 0.0446·34-s − 0.529·35-s − 0.973·37-s − 0.175·38-s − 0.343·40-s − 0.694·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.3588697840\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3588697840\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 + 2.05e5T \) |
| good | 2 | \( 1 - 7.17T + 128T^{2} \) |
| 5 | \( 1 - 94.6T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.41e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 467.T + 1.94e7T^{2} \) |
| 13 | \( 1 + 9.66e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.42e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 8.25e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 5.57e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.51e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 4.38e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.99e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 3.06e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 1.30e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 2.60e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.93e6T + 1.17e12T^{2} \) |
| 61 | \( 1 - 1.26e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 5.44e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.13e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.81e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 5.63e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 4.91e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.04e7T + 4.42e13T^{2} \) |
| 97 | \( 1 + 5.40e5T + 8.07e13T^{2} \) |
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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
−9.648703722265188441306844836785, −9.144471845244926641924668474961, −7.87361302061155658823941019238, −6.72222894102216777376411353548, −5.95029338495818323951717376439, −5.10806725628513127137405892724, −3.94846401335128955146410851094, −3.21924006265628139017662212527, −2.07722889861512393099767755700, −0.22079956587764077071055880963,
0.22079956587764077071055880963, 2.07722889861512393099767755700, 3.21924006265628139017662212527, 3.94846401335128955146410851094, 5.10806725628513127137405892724, 5.95029338495818323951717376439, 6.72222894102216777376411353548, 7.87361302061155658823941019238, 9.144471845244926641924668474961, 9.648703722265188441306844836785
