| L(s) = 1 | + 6.12·3-s + 59.9·5-s − 44.7·7-s − 205.·9-s − 774.·11-s − 629.·13-s + 367.·15-s + 1.62e3·17-s − 361·19-s − 274.·21-s − 746.·23-s + 469.·25-s − 2.74e3·27-s − 5.39e3·29-s + 2.23e3·31-s − 4.74e3·33-s − 2.68e3·35-s + 1.28e4·37-s − 3.85e3·39-s − 1.60e4·41-s − 2.06e4·43-s − 1.23e4·45-s + 1.11e4·47-s − 1.48e4·49-s + 9.96e3·51-s + 1.40e4·53-s − 4.64e4·55-s + ⋯ |
| L(s) = 1 | + 0.393·3-s + 1.07·5-s − 0.344·7-s − 0.845·9-s − 1.93·11-s − 1.03·13-s + 0.421·15-s + 1.36·17-s − 0.229·19-s − 0.135·21-s − 0.294·23-s + 0.150·25-s − 0.725·27-s − 1.19·29-s + 0.417·31-s − 0.758·33-s − 0.369·35-s + 1.54·37-s − 0.405·39-s − 1.48·41-s − 1.70·43-s − 0.906·45-s + 0.736·47-s − 0.880·49-s + 0.536·51-s + 0.686·53-s − 2.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + 361T \) |
| good | 3 | \( 1 - 6.12T + 243T^{2} \) |
| 5 | \( 1 - 59.9T + 3.12e3T^{2} \) |
| 7 | \( 1 + 44.7T + 1.68e4T^{2} \) |
| 11 | \( 1 + 774.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 629.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.62e3T + 1.41e6T^{2} \) |
| 23 | \( 1 + 746.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.39e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.23e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.28e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.60e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.06e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.11e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.40e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.30e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.41e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.20e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 7.07e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.04e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.68e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.67e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.68e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 9.14e4T + 8.58e9T^{2} \) |
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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
−11.59826719184686997731404386317, −10.14327558185593094648387758102, −9.820977888484780787669258554874, −8.396192366002420537485458046614, −7.47775634294480312156469607851, −5.86422046643037772244462268677, −5.18163226656818785354470659015, −3.08286810983575809321366436243, −2.17593143770389040882512909095, 0,
2.17593143770389040882512909095, 3.08286810983575809321366436243, 5.18163226656818785354470659015, 5.86422046643037772244462268677, 7.47775634294480312156469607851, 8.396192366002420537485458046614, 9.820977888484780787669258554874, 10.14327558185593094648387758102, 11.59826719184686997731404386317
