| L(s) = 1 | + 1.13i·2-s − 1.10i·3-s + 0.714·4-s + 1.25·6-s − 2.46i·7-s + 3.07i·8-s + 1.77·9-s + 1.71·11-s − 0.790i·12-s + 6.49i·13-s + 2.79·14-s − 2.05·16-s − 3.32i·17-s + 2.01i·18-s − 0.734·19-s + ⋯ |
| L(s) = 1 | + 0.801i·2-s − 0.638i·3-s + 0.357·4-s + 0.511·6-s − 0.930i·7-s + 1.08i·8-s + 0.592·9-s + 0.517·11-s − 0.228i·12-s + 1.80i·13-s + 0.745·14-s − 0.514·16-s − 0.807i·17-s + 0.474i·18-s − 0.168·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.98684 + 0.469029i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.98684 + 0.469029i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 37 | \( 1 + iT \) |
| good | 2 | \( 1 - 1.13iT - 2T^{2} \) |
| 3 | \( 1 + 1.10iT - 3T^{2} \) |
| 7 | \( 1 + 2.46iT - 7T^{2} \) |
| 11 | \( 1 - 1.71T + 11T^{2} \) |
| 13 | \( 1 - 6.49iT - 13T^{2} \) |
| 17 | \( 1 + 3.32iT - 17T^{2} \) |
| 19 | \( 1 + 0.734T + 19T^{2} \) |
| 23 | \( 1 + 2.08iT - 23T^{2} \) |
| 29 | \( 1 - 4.21T + 29T^{2} \) |
| 31 | \( 1 - 7.46T + 31T^{2} \) |
| 41 | \( 1 - 1.71T + 41T^{2} \) |
| 43 | \( 1 - 1.81iT - 43T^{2} \) |
| 47 | \( 1 - 0.882iT - 47T^{2} \) |
| 53 | \( 1 + 7.03iT - 53T^{2} \) |
| 59 | \( 1 + 0.387T + 59T^{2} \) |
| 61 | \( 1 + 11.8T + 61T^{2} \) |
| 67 | \( 1 + 12.1iT - 67T^{2} \) |
| 71 | \( 1 - 13.7T + 71T^{2} \) |
| 73 | \( 1 - 16.6iT - 73T^{2} \) |
| 79 | \( 1 + 8.23T + 79T^{2} \) |
| 83 | \( 1 + 4.80iT - 83T^{2} \) |
| 89 | \( 1 - 1.52T + 89T^{2} \) |
| 97 | \( 1 - 18.1iT - 97T^{2} \) |
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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
−10.07488023725122835997391684888, −9.169291527435191184066697522640, −8.142352151831403511885983370243, −7.36722947458925537913591585589, −6.61988348170547232899469537456, −6.47607747157856842710924483863, −4.84906495161139323954419046204, −4.09043695478173306225715735418, −2.43339479618237584423936678547, −1.27574617689146613673240302989,
1.25853469616549786254046205816, 2.61833789130216944471228008977, 3.44421691079960881091098306510, 4.46798749875885613636672482166, 5.63689643775260756779271072822, 6.43185584820244402430698379754, 7.58006984023530570711213831218, 8.520902125463964422718479580700, 9.470075660080596806431868636329, 10.25916090895516801687648164924
