| L(s) = 1 | + 1.83i·2-s − 11.9i·3-s + 28.6·4-s + 21.9·6-s + 112. i·7-s + 111. i·8-s + 100.·9-s + 162.·11-s − 341. i·12-s + 169i·13-s − 206.·14-s + 711.·16-s + 379. i·17-s + 185. i·18-s + 284.·19-s + ⋯ |
| L(s) = 1 | + 0.324i·2-s − 0.765i·3-s + 0.894·4-s + 0.248·6-s + 0.865i·7-s + 0.615i·8-s + 0.414·9-s + 0.405·11-s − 0.684i·12-s + 0.277i·13-s − 0.281·14-s + 0.694·16-s + 0.318i·17-s + 0.134i·18-s + 0.180·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.695389969\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.695389969\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 - 169iT \) |
| good | 2 | \( 1 - 1.83iT - 32T^{2} \) |
| 3 | \( 1 + 11.9iT - 243T^{2} \) |
| 7 | \( 1 - 112. iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 162.T + 1.61e5T^{2} \) |
| 17 | \( 1 - 379. iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 284.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.18e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 6.27e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.55e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.06e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 4.16e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.15e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 2.26e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 4.77e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 4.83e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.65e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.58e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 3.72e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.93e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 4.49e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.37e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 561.T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.12e4iT - 8.58e9T^{2} \) |
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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
−11.30280323798401902851740368916, −9.951403606597783620156376720167, −8.943117663102984580528734133793, −7.76688859930420890803411724263, −7.14808661888949397527802209113, −6.18864393037342419137371164722, −5.37621896241210634047300455149, −3.61534640569278987666313497730, −2.17912972510250573612173677230, −1.41641279620037327964612203024,
0.68573794385259305979094688765, 2.05023243237425102238506974366, 3.50124925730503693395997566675, 4.24468832922379269888108130631, 5.62021496414843989268539560769, 6.92560150943966331770121468514, 7.49427588879352611399086282803, 8.993475696879959283018405981942, 9.952694662998735051047687477136, 10.67024502496290920114450006736
