| L(s) = 1 | − 0.894·2-s − 1.19·4-s − 3.74·5-s − 1.45·7-s + 2.86·8-s + 3.35·10-s − 5.75·13-s + 1.29·14-s − 0.160·16-s − 4.79·17-s − 0.702·19-s + 4.49·20-s + 1.65·23-s + 9.05·25-s + 5.14·26-s + 1.74·28-s − 4.30·29-s − 3.30·31-s − 5.58·32-s + 4.29·34-s + 5.43·35-s + 9.73·37-s + 0.628·38-s − 10.7·40-s + 4.24·41-s + 4.10·43-s − 1.47·46-s + ⋯ |
| L(s) = 1 | − 0.632·2-s − 0.599·4-s − 1.67·5-s − 0.548·7-s + 1.01·8-s + 1.06·10-s − 1.59·13-s + 0.346·14-s − 0.0400·16-s − 1.16·17-s − 0.161·19-s + 1.00·20-s + 0.344·23-s + 1.81·25-s + 1.00·26-s + 0.329·28-s − 0.799·29-s − 0.593·31-s − 0.986·32-s + 0.735·34-s + 0.919·35-s + 1.60·37-s + 0.101·38-s − 1.69·40-s + 0.663·41-s + 0.626·43-s − 0.217·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9801 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9801 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 0.894T + 2T^{2} \) |
| 5 | \( 1 + 3.74T + 5T^{2} \) |
| 7 | \( 1 + 1.45T + 7T^{2} \) |
| 13 | \( 1 + 5.75T + 13T^{2} \) |
| 17 | \( 1 + 4.79T + 17T^{2} \) |
| 19 | \( 1 + 0.702T + 19T^{2} \) |
| 23 | \( 1 - 1.65T + 23T^{2} \) |
| 29 | \( 1 + 4.30T + 29T^{2} \) |
| 31 | \( 1 + 3.30T + 31T^{2} \) |
| 37 | \( 1 - 9.73T + 37T^{2} \) |
| 41 | \( 1 - 4.24T + 41T^{2} \) |
| 43 | \( 1 - 4.10T + 43T^{2} \) |
| 47 | \( 1 + 1.79T + 47T^{2} \) |
| 53 | \( 1 - 1.15T + 53T^{2} \) |
| 59 | \( 1 - 4.65T + 59T^{2} \) |
| 61 | \( 1 + 2.54T + 61T^{2} \) |
| 67 | \( 1 - 8.94T + 67T^{2} \) |
| 71 | \( 1 + 5.14T + 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 - 1.08T + 79T^{2} \) |
| 83 | \( 1 + 3.80T + 83T^{2} \) |
| 89 | \( 1 - 4.01T + 89T^{2} \) |
| 97 | \( 1 - 3.29T + 97T^{2} \) |
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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
−7.32603831883172418387742839761, −7.16383993791885457600022626701, −6.06576112036461783569007053082, −4.97704487914574201717148610936, −4.48045233888696111882649850904, −3.95591606077837410870886894647, −3.09532975955575089851345604282, −2.15022686455453595394042550785, −0.67236455406071676145982277962, 0,
0.67236455406071676145982277962, 2.15022686455453595394042550785, 3.09532975955575089851345604282, 3.95591606077837410870886894647, 4.48045233888696111882649850904, 4.97704487914574201717148610936, 6.06576112036461783569007053082, 7.16383993791885457600022626701, 7.32603831883172418387742839761
