| L(s) = 1 | + 2.20·3-s + (−1.71 − 1.43i)5-s − 0.642i·7-s + 1.85·9-s − 2.87·11-s − 0.405i·13-s + (−3.78 − 3.15i)15-s − 4.93·17-s − 2.74·19-s − 1.41i·21-s + (−4.71 − 0.896i)23-s + (0.900 + 4.91i)25-s − 2.52·27-s − 1.62·29-s − 4.55i·31-s + ⋯ |
| L(s) = 1 | + 1.27·3-s + (−0.768 − 0.640i)5-s − 0.242i·7-s + 0.618·9-s − 0.865·11-s − 0.112i·13-s + (−0.977 − 0.814i)15-s − 1.19·17-s − 0.629·19-s − 0.308i·21-s + (−0.982 − 0.186i)23-s + (0.180 + 0.983i)25-s − 0.485·27-s − 0.302·29-s − 0.818i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 + 0.163i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.986 + 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5785254139\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5785254139\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.71 + 1.43i)T \) |
| 23 | \( 1 + (4.71 + 0.896i)T \) |
| good | 3 | \( 1 - 2.20T + 3T^{2} \) |
| 7 | \( 1 + 0.642iT - 7T^{2} \) |
| 11 | \( 1 + 2.87T + 11T^{2} \) |
| 13 | \( 1 + 0.405iT - 13T^{2} \) |
| 17 | \( 1 + 4.93T + 17T^{2} \) |
| 19 | \( 1 + 2.74T + 19T^{2} \) |
| 29 | \( 1 + 1.62T + 29T^{2} \) |
| 31 | \( 1 + 4.55iT - 31T^{2} \) |
| 37 | \( 1 - 2.00T + 37T^{2} \) |
| 41 | \( 1 + 4.79T + 41T^{2} \) |
| 43 | \( 1 - 3.29iT - 43T^{2} \) |
| 47 | \( 1 - 3.58T + 47T^{2} \) |
| 53 | \( 1 + 0.263T + 53T^{2} \) |
| 59 | \( 1 - 10.3iT - 59T^{2} \) |
| 61 | \( 1 + 9.19iT - 61T^{2} \) |
| 67 | \( 1 + 2.18iT - 67T^{2} \) |
| 71 | \( 1 + 10.3iT - 71T^{2} \) |
| 73 | \( 1 + 4.43iT - 73T^{2} \) |
| 79 | \( 1 - 3.16T + 79T^{2} \) |
| 83 | \( 1 - 9.14iT - 83T^{2} \) |
| 89 | \( 1 + 14.0iT - 89T^{2} \) |
| 97 | \( 1 + 10.3T + 97T^{2} \) |
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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
−8.825387062107811612181371050382, −7.985702970155366380322742574572, −7.78704454466010452851710953895, −6.69032619044175416058056812970, −5.54281612949648019459725931535, −4.43521666420419347813752047909, −3.90590621091055612089360554422, −2.82065804418593807332438299316, −1.94397352808334558741543409027, −0.16071499251518160426768289128,
2.11787440940980216635347243346, 2.72034621539969458590251417895, 3.68708401198931267712234824876, 4.38976117606937716709220314046, 5.62092336618042471881872498910, 6.73055693287285509378702547998, 7.38832177422160019303003777909, 8.282213985613029012184759194255, 8.535073931224523460801640216092, 9.480585067779114258770624501490
