| L(s) = 1 | + 30.5i·3-s + 4.01e3i·7-s + 1.87e4·9-s + 4.21e4·11-s − 1.23e5i·13-s + 3.19e5i·17-s + 1.08e6·19-s − 1.22e5·21-s + 1.50e6i·23-s + 1.17e6i·27-s + 2.62e6·29-s − 3.27e6·31-s + 1.28e6i·33-s − 2.51e6i·37-s + 3.77e6·39-s + ⋯ |
| L(s) = 1 | + 0.217i·3-s + 0.631i·7-s + 0.952·9-s + 0.867·11-s − 1.20i·13-s + 0.929i·17-s + 1.91·19-s − 0.137·21-s + 1.12i·23-s + 0.424i·27-s + 0.688·29-s − 0.635·31-s + 0.188i·33-s − 0.220i·37-s + 0.261·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(3.208193822\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.208193822\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 30.5iT - 1.96e4T^{2} \) |
| 7 | \( 1 - 4.01e3iT - 4.03e7T^{2} \) |
| 11 | \( 1 - 4.21e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.23e5iT - 1.06e10T^{2} \) |
| 17 | \( 1 - 3.19e5iT - 1.18e11T^{2} \) |
| 19 | \( 1 - 1.08e6T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.50e6iT - 1.80e12T^{2} \) |
| 29 | \( 1 - 2.62e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 3.27e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 2.51e6iT - 1.29e14T^{2} \) |
| 41 | \( 1 - 2.95e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.42e7iT - 5.02e14T^{2} \) |
| 47 | \( 1 + 1.35e6iT - 1.11e15T^{2} \) |
| 53 | \( 1 + 9.73e7iT - 3.29e15T^{2} \) |
| 59 | \( 1 + 7.48e6T + 8.66e15T^{2} \) |
| 61 | \( 1 + 9.11e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.94e8iT - 2.72e16T^{2} \) |
| 71 | \( 1 + 1.56e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.82e8iT - 5.88e16T^{2} \) |
| 79 | \( 1 + 5.55e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 6.48e6iT - 1.86e17T^{2} \) |
| 89 | \( 1 - 5.99e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 9.25e8iT - 7.60e17T^{2} \) |
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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
−9.746259270151261787441310749041, −9.101103530276393220667270011026, −7.928538943230405360090802258514, −7.16210247098252063848997919689, −5.92775445156478555472661066563, −5.19927767040135051932822659247, −3.93454477956409136689438397863, −3.12461807212778639902600656244, −1.70898977459799241955343976111, −0.835762965368064448110009432000,
0.810679349746540469182564517988, 1.42222287757630937338537115756, 2.78772903265664482666482944416, 4.07125541120097674689178787782, 4.68173064625099262439578799867, 6.12652797162346666844064470123, 7.11251868856772744058667984861, 7.49891478724522529987247803688, 9.013473748771229546823065070888, 9.580413058516746932471302438669
