| L(s) = 1 | − 3-s − 0.601i·5-s + 9-s − 0.163i·11-s + 3.37i·13-s + 0.601i·15-s − 2.13i·17-s − 0.953·19-s − 6.99i·23-s + 4.63·25-s − 27-s − 1.28·29-s − 3.35·31-s + 0.163i·33-s + 5.16·37-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.269i·5-s + 0.333·9-s − 0.0493i·11-s + 0.937i·13-s + 0.155i·15-s − 0.517i·17-s − 0.218·19-s − 1.45i·23-s + 0.927·25-s − 0.192·27-s − 0.239·29-s − 0.602·31-s + 0.0284i·33-s + 0.848·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.777 + 0.629i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.777 + 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.323625436\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.323625436\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 0.601iT - 5T^{2} \) |
| 11 | \( 1 + 0.163iT - 11T^{2} \) |
| 13 | \( 1 - 3.37iT - 13T^{2} \) |
| 17 | \( 1 + 2.13iT - 17T^{2} \) |
| 19 | \( 1 + 0.953T + 19T^{2} \) |
| 23 | \( 1 + 6.99iT - 23T^{2} \) |
| 29 | \( 1 + 1.28T + 29T^{2} \) |
| 31 | \( 1 + 3.35T + 31T^{2} \) |
| 37 | \( 1 - 5.16T + 37T^{2} \) |
| 41 | \( 1 - 6.67iT - 41T^{2} \) |
| 43 | \( 1 - 6.09iT - 43T^{2} \) |
| 47 | \( 1 - 3.69T + 47T^{2} \) |
| 53 | \( 1 - 13.8T + 53T^{2} \) |
| 59 | \( 1 + 8.05T + 59T^{2} \) |
| 61 | \( 1 - 0.181iT - 61T^{2} \) |
| 67 | \( 1 + 9.09iT - 67T^{2} \) |
| 71 | \( 1 + 3.36iT - 71T^{2} \) |
| 73 | \( 1 + 12.1iT - 73T^{2} \) |
| 79 | \( 1 + 6.98iT - 79T^{2} \) |
| 83 | \( 1 - 12.9T + 83T^{2} \) |
| 89 | \( 1 + 14.8iT - 89T^{2} \) |
| 97 | \( 1 - 9.08iT - 97T^{2} \) |
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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
−9.015084738485289974163209457917, −8.156046570171394575971691444941, −7.23388352055262179918598107761, −6.53556481668307500858327717906, −5.85588821835996211196250985182, −4.74382353814731147476416068095, −4.38924176554797887824198771449, −3.07766401779103361979065622440, −1.93802734994769447247813435635, −0.63987503413117981143280516090,
0.928764644430863002637984163523, 2.24717047426697890565123901745, 3.40332550553549439170025268355, 4.20224118607877070635057468640, 5.44250506290614248638033313147, 5.68827061640753108523316180660, 6.83865922395364411551676760072, 7.38995667060575480419473168774, 8.257773901687760283537083912471, 9.096195334381683752984869527725
