| L(s) = 1 | − 3-s + 3.46i·7-s − 2·9-s + (−1 − 3.46i)13-s − 6·17-s − 6.92i·19-s − 3.46i·21-s + 3·23-s + 5·27-s + 9·29-s − 3.46i·31-s + 10.3i·37-s + (1 + 3.46i)39-s − 10.3i·41-s + 5·43-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.30i·7-s − 0.666·9-s + (−0.277 − 0.960i)13-s − 1.45·17-s − 1.58i·19-s − 0.755i·21-s + 0.625·23-s + 0.962·27-s + 1.67·29-s − 0.622i·31-s + 1.70i·37-s + (0.160 + 0.554i)39-s − 1.62i·41-s + 0.762·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7995803231\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7995803231\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (1 + 3.46i)T \) |
| good | 3 | \( 1 + T + 3T^{2} \) |
| 7 | \( 1 - 3.46iT - 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 + 6.92iT - 19T^{2} \) |
| 23 | \( 1 - 3T + 23T^{2} \) |
| 29 | \( 1 - 9T + 29T^{2} \) |
| 31 | \( 1 + 3.46iT - 31T^{2} \) |
| 37 | \( 1 - 10.3iT - 37T^{2} \) |
| 41 | \( 1 + 10.3iT - 41T^{2} \) |
| 43 | \( 1 - 5T + 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 3T + 53T^{2} \) |
| 59 | \( 1 + 10.3iT - 59T^{2} \) |
| 61 | \( 1 + 5T + 61T^{2} \) |
| 67 | \( 1 + 6.92iT - 67T^{2} \) |
| 71 | \( 1 + 10.3iT - 71T^{2} \) |
| 73 | \( 1 + 10.3iT - 73T^{2} \) |
| 79 | \( 1 - T + 79T^{2} \) |
| 83 | \( 1 + 10.3iT - 83T^{2} \) |
| 89 | \( 1 - 10.3iT - 89T^{2} \) |
| 97 | \( 1 + 3.46iT - 97T^{2} \) |
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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
−9.225850340894946771867494557987, −8.826616744938268247511877166818, −8.013576163511104118151644281056, −6.74992009478053150045173356907, −6.18613310039803023432596297801, −5.19453695894937908673653389420, −4.72991393456673997217943883188, −2.99173276423038915237806146469, −2.39619370547679499953953692783, −0.40072598403499828659890117687,
1.14776330815722045036744224108, 2.63686310282977883343882505651, 4.00547478389293069716635725533, 4.57819666883228889991181356074, 5.71663434452363801104564006351, 6.61319538149309746253071293062, 7.15515301205916312025858267957, 8.226414670766457466352415160788, 8.977905919922251236535985741029, 9.988845125209807975458147135807
