| L(s) = 1 | + (−0.139 + 0.967i)2-s + (−0.415 − 0.909i)3-s + (1.00 + 0.294i)4-s + (0.654 + 0.755i)5-s + (0.937 − 0.275i)6-s + (1.72 + 1.10i)7-s + (−1.23 + 2.70i)8-s + (−0.654 + 0.755i)9-s + (−0.822 + 0.528i)10-s + (−0.270 − 1.88i)11-s + (−0.148 − 1.03i)12-s + (0.240 − 0.154i)13-s + (−1.30 + 1.51i)14-s + (0.415 − 0.909i)15-s + (−0.687 − 0.441i)16-s + (−0.126 + 0.0372i)17-s + ⋯ |
| L(s) = 1 | + (−0.0983 + 0.683i)2-s + (−0.239 − 0.525i)3-s + (0.501 + 0.147i)4-s + (0.292 + 0.337i)5-s + (0.382 − 0.112i)6-s + (0.650 + 0.417i)7-s + (−0.437 + 0.956i)8-s + (−0.218 + 0.251i)9-s + (−0.259 + 0.167i)10-s + (−0.0816 − 0.567i)11-s + (−0.0429 − 0.298i)12-s + (0.0666 − 0.0428i)13-s + (−0.349 + 0.403i)14-s + (0.107 − 0.234i)15-s + (−0.171 − 0.110i)16-s + (−0.0307 + 0.00902i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.445 - 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.445 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.26554 + 0.783594i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.26554 + 0.783594i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.415 + 0.909i)T \) |
| 5 | \( 1 + (-0.654 - 0.755i)T \) |
| 23 | \( 1 + (2.29 - 4.21i)T \) |
| good | 2 | \( 1 + (0.139 - 0.967i)T + (-1.91 - 0.563i)T^{2} \) |
| 7 | \( 1 + (-1.72 - 1.10i)T + (2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (0.270 + 1.88i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (-0.240 + 0.154i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (0.126 - 0.0372i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (-5.40 - 1.58i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (-3.25 + 0.954i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-0.903 + 1.97i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (1.01 - 1.17i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (4.25 + 4.90i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-0.224 - 0.491i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 - 1.57T + 47T^{2} \) |
| 53 | \( 1 + (5.99 + 3.85i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (-1.49 + 0.963i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-2.99 + 6.55i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (-1.59 + 11.1i)T + (-64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (-0.764 + 5.31i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (2.58 + 0.758i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (3.30 - 2.12i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (10.0 - 11.6i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (0.797 + 1.74i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (9.45 + 10.9i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−11.61047216357181028901859967884, −11.04237650445188144127147001741, −9.785445154896639496937392198934, −8.472232764559343432125405919621, −7.81905433148069409650345447855, −6.90252913910274821379601191771, −5.92768266418816844865090443876, −5.24536372104767751217626717664, −3.19204022765799976616247591755, −1.84438896406678355317412488890,
1.28545912124773760900627076468, 2.78011375457143343673110322048, 4.22241952558229784066752653474, 5.24962790056766802168850065418, 6.47998011064818835843950320954, 7.53283139815764569097893101669, 8.815401778910735635740483718739, 9.876951207298457434470130562796, 10.38454736064891422069686097647, 11.33874389677152767730971398237
