| L(s) = 1 | − 27·3-s − 129.·5-s + 729·9-s − 4.51e3·11-s + 7.88e3·13-s + 3.49e3·15-s − 7.08e3·17-s − 1.40e4·19-s + 6.90e4·23-s − 6.13e4·25-s − 1.96e4·27-s + 4.73e4·29-s − 1.70e5·31-s + 1.22e5·33-s + 2.66e5·37-s − 2.13e5·39-s + 5.74e5·41-s + 1.09e4·43-s − 9.44e4·45-s + 1.53e5·47-s + 1.91e5·51-s − 1.66e6·53-s + 5.85e5·55-s + 3.79e5·57-s + 4.11e5·59-s + 2.67e6·61-s − 1.02e6·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.463·5-s + 0.333·9-s − 1.02·11-s + 0.995·13-s + 0.267·15-s − 0.349·17-s − 0.470·19-s + 1.18·23-s − 0.785·25-s − 0.192·27-s + 0.360·29-s − 1.02·31-s + 0.591·33-s + 0.865·37-s − 0.575·39-s + 1.30·41-s + 0.0209·43-s − 0.154·45-s + 0.216·47-s + 0.201·51-s − 1.53·53-s + 0.474·55-s + 0.271·57-s + 0.261·59-s + 1.50·61-s − 0.461·65-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
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 + 129.T + 7.81e4T^{2} \) |
| 11 | \( 1 + 4.51e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 7.88e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 7.08e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.40e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 6.90e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 4.73e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.70e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.66e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 5.74e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 1.09e4T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.53e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.66e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 4.11e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.67e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.11e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 5.51e4T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.15e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 4.28e5T + 1.92e13T^{2} \) |
| 83 | \( 1 + 7.54e5T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.18e7T + 4.42e13T^{2} \) |
| 97 | \( 1 - 7.12e6T + 8.07e13T^{2} \) |
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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
−9.107029320786189168271238654177, −8.171759338985244130123338797068, −7.37735289603475483950604511053, −6.36131680495177879727323686950, −5.50351123279456220994426351622, −4.53451548562499711221742494579, −3.55830368445329126173359540175, −2.33130896405830937589870748590, −0.999468640542885247073741781067, 0,
0.999468640542885247073741781067, 2.33130896405830937589870748590, 3.55830368445329126173359540175, 4.53451548562499711221742494579, 5.50351123279456220994426351622, 6.36131680495177879727323686950, 7.37735289603475483950604511053, 8.171759338985244130123338797068, 9.107029320786189168271238654177
