| L(s) = 1 | + 2.36i·2-s + i·3-s − 3.58·4-s + (0.979 − 2.01i)5-s − 2.36·6-s − 1.71i·7-s − 3.75i·8-s − 9-s + (4.75 + 2.31i)10-s − 3.58i·12-s + 3.53i·13-s + 4.05·14-s + (2.01 + 0.979i)15-s + 1.69·16-s − 0.373i·17-s − 2.36i·18-s + ⋯ |
| L(s) = 1 | + 1.67i·2-s + 0.577i·3-s − 1.79·4-s + (0.437 − 0.899i)5-s − 0.965·6-s − 0.647i·7-s − 1.32i·8-s − 0.333·9-s + (1.50 + 0.731i)10-s − 1.03i·12-s + 0.979i·13-s + 1.08·14-s + (0.519 + 0.252i)15-s + 0.423·16-s − 0.0905i·17-s − 0.557i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 + 0.437i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.899 + 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4601668698\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4601668698\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - iT \) |
| 5 | \( 1 + (-0.979 + 2.01i)T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.36iT - 2T^{2} \) |
| 7 | \( 1 + 1.71iT - 7T^{2} \) |
| 13 | \( 1 - 3.53iT - 13T^{2} \) |
| 17 | \( 1 + 0.373iT - 17T^{2} \) |
| 19 | \( 1 + 5.76T + 19T^{2} \) |
| 23 | \( 1 + 3.80iT - 23T^{2} \) |
| 29 | \( 1 + 3.51T + 29T^{2} \) |
| 31 | \( 1 + 10.0T + 31T^{2} \) |
| 37 | \( 1 + 11.5iT - 37T^{2} \) |
| 41 | \( 1 - 2.41T + 41T^{2} \) |
| 43 | \( 1 - 0.224iT - 43T^{2} \) |
| 47 | \( 1 - 4.67iT - 47T^{2} \) |
| 53 | \( 1 + 2.44iT - 53T^{2} \) |
| 59 | \( 1 + 5.35T + 59T^{2} \) |
| 61 | \( 1 - 3.40T + 61T^{2} \) |
| 67 | \( 1 + 9.08iT - 67T^{2} \) |
| 71 | \( 1 + 8.77T + 71T^{2} \) |
| 73 | \( 1 - 14.9iT - 73T^{2} \) |
| 79 | \( 1 - 8.79T + 79T^{2} \) |
| 83 | \( 1 + 7.44iT - 83T^{2} \) |
| 89 | \( 1 + 15.5T + 89T^{2} \) |
| 97 | \( 1 + 14.6iT - 97T^{2} \) |
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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
−9.077144559202636403235350366019, −8.448433431038268648461118129346, −7.54463291308752893005425257654, −6.81521038454538666788229895799, −5.96206284071838661685621009478, −5.34317216575473169545493451178, −4.32975471602877604189356957737, −4.07544869863297330975987853923, −2.03500200323322413150420466706, −0.16228900812547308934506588494,
1.53683643428697324876941226518, 2.32332829600759946663440431797, 3.06456050368848974725050392696, 3.87661618674205706987357747737, 5.23457015096837539559137729653, 5.97866788698525866242701397837, 6.94922497110939552895599348183, 7.926005829740953322060672646579, 8.841647134764196761062292332658, 9.501748077682863677435361915245
