| L(s) = 1 | + (0.809 + 0.587i)2-s + (−0.913 − 0.406i)3-s + (0.309 + 0.951i)4-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)6-s + (0.334 − 0.370i)7-s + (−0.309 + 0.951i)8-s + (0.669 + 0.743i)9-s + (−0.913 + 0.406i)10-s + (1.48 + 0.315i)11-s + (0.104 − 0.994i)12-s + (0.0540 + 0.513i)13-s + (0.488 − 0.103i)14-s + (0.809 − 0.587i)15-s + (−0.809 + 0.587i)16-s + (−1.01 + 0.215i)17-s + ⋯ |
| L(s) = 1 | + (0.572 + 0.415i)2-s + (−0.527 − 0.234i)3-s + (0.154 + 0.475i)4-s + (−0.223 + 0.387i)5-s + (−0.204 − 0.353i)6-s + (0.126 − 0.140i)7-s + (−0.109 + 0.336i)8-s + (0.223 + 0.247i)9-s + (−0.288 + 0.128i)10-s + (0.447 + 0.0950i)11-s + (0.0301 − 0.287i)12-s + (0.0149 + 0.142i)13-s + (0.130 − 0.0277i)14-s + (0.208 − 0.151i)15-s + (−0.202 + 0.146i)16-s + (−0.246 + 0.0523i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.264 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.264 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.941883 + 1.23474i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.941883 + 1.23474i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.809 - 0.587i)T \) |
| 3 | \( 1 + (0.913 + 0.406i)T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 31 | \( 1 + (-5.07 + 2.28i)T \) |
| good | 7 | \( 1 + (-0.334 + 0.370i)T + (-0.731 - 6.96i)T^{2} \) |
| 11 | \( 1 + (-1.48 - 0.315i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (-0.0540 - 0.513i)T + (-12.7 + 2.70i)T^{2} \) |
| 17 | \( 1 + (1.01 - 0.215i)T + (15.5 - 6.91i)T^{2} \) |
| 19 | \( 1 + (0.712 - 6.78i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (1.07 - 3.30i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-3.15 - 2.28i)T + (8.96 + 27.5i)T^{2} \) |
| 37 | \( 1 + (-2.95 - 5.11i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (9.93 - 4.42i)T + (27.4 - 30.4i)T^{2} \) |
| 43 | \( 1 + (0.0609 - 0.580i)T + (-42.0 - 8.94i)T^{2} \) |
| 47 | \( 1 + (7.33 - 5.32i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (3.19 + 3.54i)T + (-5.54 + 52.7i)T^{2} \) |
| 59 | \( 1 + (-6.42 - 2.86i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 + 5.45T + 61T^{2} \) |
| 67 | \( 1 + (-4.91 + 8.51i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.38 - 1.53i)T + (-7.42 + 70.6i)T^{2} \) |
| 73 | \( 1 + (-0.946 - 0.201i)T + (66.6 + 29.6i)T^{2} \) |
| 79 | \( 1 + (-12.4 + 2.63i)T + (72.1 - 32.1i)T^{2} \) |
| 83 | \( 1 + (-2.58 + 1.14i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + (-2.80 - 8.63i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (0.749 + 2.30i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−10.39341865533474499224342962986, −9.600947832371044225970109528846, −8.259563903140856115741328758732, −7.73829916912719350080962737033, −6.60338726336296692704086082500, −6.22688343597841155689043829918, −5.07752484122878260159063773828, −4.18176797738401206347678621904, −3.17578705255519799026344015021, −1.61901753821690449575063687872,
0.66874617153973132182796495147, 2.27801097653833802063160158156, 3.57372714047492017772739292280, 4.60545916833502932039966336468, 5.14943331413210955086785928126, 6.31356524293080035851544393691, 6.96223237981870588105575904123, 8.317704285264046737016507418425, 9.067389230486056979808729134478, 10.02127304462318101478537781720
