L(s) = 1 | + (−0.382 + 0.923i)2-s + (−0.707 − 0.707i)4-s + (−0.216 − 1.08i)5-s + (0.923 − 0.382i)8-s + (−0.382 − 0.923i)9-s + (1.08 + 0.216i)10-s + (1 + i)13-s + i·16-s + (0.923 + 0.382i)17-s + 18-s + (−0.617 + 0.923i)20-s + (−0.216 + 0.0897i)25-s + (−1.30 + 0.541i)26-s + (−0.216 − 1.08i)29-s + (−0.923 − 0.382i)32-s + ⋯ |
L(s) = 1 | + (−0.382 + 0.923i)2-s + (−0.707 − 0.707i)4-s + (−0.216 − 1.08i)5-s + (0.923 − 0.382i)8-s + (−0.382 − 0.923i)9-s + (1.08 + 0.216i)10-s + (1 + i)13-s + i·16-s + (0.923 + 0.382i)17-s + 18-s + (−0.617 + 0.923i)20-s + (−0.216 + 0.0897i)25-s + (−1.30 + 0.541i)26-s + (−0.216 − 1.08i)29-s + (−0.923 − 0.382i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 + 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 + 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8953273090\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8953273090\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.382 - 0.923i)T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + (-0.923 - 0.382i)T \) |
good | 3 | \( 1 + (0.382 + 0.923i)T^{2} \) |
| 5 | \( 1 + (0.216 + 1.08i)T + (-0.923 + 0.382i)T^{2} \) |
| 11 | \( 1 + (0.382 - 0.923i)T^{2} \) |
| 13 | \( 1 + (-1 - i)T + iT^{2} \) |
| 19 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 23 | \( 1 + (-0.382 + 0.923i)T^{2} \) |
| 29 | \( 1 + (0.216 + 1.08i)T + (-0.923 + 0.382i)T^{2} \) |
| 31 | \( 1 + (-0.382 - 0.923i)T^{2} \) |
| 37 | \( 1 + (1.38 + 0.923i)T + (0.382 + 0.923i)T^{2} \) |
| 41 | \( 1 + (0.0761 - 0.382i)T + (-0.923 - 0.382i)T^{2} \) |
| 43 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 + (-0.707 + 1.70i)T + (-0.707 - 0.707i)T^{2} \) |
| 59 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 61 | \( 1 + (-1.92 - 0.382i)T + (0.923 + 0.382i)T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + (-0.382 - 0.923i)T^{2} \) |
| 73 | \( 1 + (0.324 + 1.63i)T + (-0.923 + 0.382i)T^{2} \) |
| 79 | \( 1 + (0.382 - 0.923i)T^{2} \) |
| 83 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 89 | \( 1 - iT^{2} \) |
| 97 | \( 1 + (-1.63 + 0.324i)T + (0.923 - 0.382i)T^{2} \) |
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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
−8.528144162764836409109935635184, −8.325567376290069341604780419979, −7.24832499680245111574646148901, −6.49890227374231967441727909696, −5.81180961598542003424313148672, −5.14965572982992226206634568107, −4.14603782528158693668835749806, −3.61794003478538298139978442845, −1.74653701624830739458881487962, −0.71497100138317770462887077094,
1.26756853036687390816349607319, 2.52400796067588744735983381558, 3.17918083295035340943181955500, 3.78175941483594463687066918970, 5.06123448042823153771840038261, 5.65131207676886235115997666376, 6.89426830854985449723079035976, 7.52028103317021682609282462945, 8.262509348291395741595436088243, 8.766171725077215665617154938091