| L(s) = 1 | + 3-s + (−1.22 + 1.87i)5-s − 2·9-s + 4.89·11-s + 4.58i·13-s + (−1.22 + 1.87i)15-s + 2.44·17-s + 2.44·19-s + (−3 + 3.74i)23-s + (−2 − 4.58i)25-s − 5·27-s − 3·29-s − 4.58i·31-s + 4.89·33-s − 4.89·37-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + (−0.547 + 0.836i)5-s − 0.666·9-s + 1.47·11-s + 1.27i·13-s + (−0.316 + 0.483i)15-s + 0.594·17-s + 0.561·19-s + (−0.625 + 0.780i)23-s + (−0.400 − 0.916i)25-s − 0.962·27-s − 0.557·29-s − 0.823i·31-s + 0.852·33-s − 0.805·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.310 - 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.310 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.561692599\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.561692599\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.22 - 1.87i)T \) |
| 23 | \( 1 + (3 - 3.74i)T \) |
| good | 3 | \( 1 - T + 3T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 4.89T + 11T^{2} \) |
| 13 | \( 1 - 4.58iT - 13T^{2} \) |
| 17 | \( 1 - 2.44T + 17T^{2} \) |
| 19 | \( 1 - 2.44T + 19T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 + 4.58iT - 31T^{2} \) |
| 37 | \( 1 + 4.89T + 37T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 - 11.2iT - 43T^{2} \) |
| 47 | \( 1 + 9T + 47T^{2} \) |
| 53 | \( 1 - 4.89T + 53T^{2} \) |
| 59 | \( 1 - 9.16iT - 59T^{2} \) |
| 61 | \( 1 - 11.2iT - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 - 13.7iT - 71T^{2} \) |
| 73 | \( 1 - 4.58iT - 73T^{2} \) |
| 79 | \( 1 - 7.34T + 79T^{2} \) |
| 83 | \( 1 + 3.74iT - 83T^{2} \) |
| 89 | \( 1 + 3.74iT - 89T^{2} \) |
| 97 | \( 1 + 9.79T + 97T^{2} \) |
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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
−9.403053071224399953500961090181, −8.760914035027131734209386337846, −7.88098445354259035754201448426, −7.17204324013359111636525405728, −6.43478046078107329586528432807, −5.61274604279707611840117975904, −4.15111639098684444103440491684, −3.68589689846598288322817894372, −2.72366215782119817803986450294, −1.54951259656422904729164029608,
0.54267309294496647913256063402, 1.83860759414197422465421681256, 3.33938658896238229278194786875, 3.71371550653991960959025391469, 4.98279313822080111030086588460, 5.65384610217344348169859750497, 6.69830845878950783420934567462, 7.70866737496961222217344883459, 8.308678797095548808501712471760, 8.894035635666149981426830181727
