| L(s) = 1 | − 2-s + 0.683·3-s + 4-s + 2.16·5-s − 0.683·6-s − 8-s − 2.53·9-s − 2.16·10-s − 3.64·11-s + 0.683·12-s − 4·13-s + 1.48·15-s + 16-s + 4.32·17-s + 2.53·18-s + 19-s + 2.16·20-s + 3.64·22-s − 1.36·23-s − 0.683·24-s − 0.316·25-s + 4·26-s − 3.78·27-s − 4.16·29-s − 1.48·30-s − 1.36·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.394·3-s + 0.5·4-s + 0.967·5-s − 0.279·6-s − 0.353·8-s − 0.844·9-s − 0.684·10-s − 1.09·11-s + 0.197·12-s − 1.10·13-s + 0.382·15-s + 0.250·16-s + 1.04·17-s + 0.596·18-s + 0.229·19-s + 0.483·20-s + 0.777·22-s − 0.285·23-s − 0.139·24-s − 0.0632·25-s + 0.784·26-s − 0.728·27-s − 0.773·29-s − 0.270·30-s − 0.245·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1862 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1862 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 0.683T + 3T^{2} \) |
| 5 | \( 1 - 2.16T + 5T^{2} \) |
| 11 | \( 1 + 3.64T + 11T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 - 4.32T + 17T^{2} \) |
| 23 | \( 1 + 1.36T + 23T^{2} \) |
| 29 | \( 1 + 4.16T + 29T^{2} \) |
| 31 | \( 1 + 1.36T + 31T^{2} \) |
| 37 | \( 1 - 0.164T + 37T^{2} \) |
| 41 | \( 1 + 11.8T + 41T^{2} \) |
| 43 | \( 1 + 7.86T + 43T^{2} \) |
| 47 | \( 1 - 6.27T + 47T^{2} \) |
| 53 | \( 1 + 1.64T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 + 2.27T + 61T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 + 6.05T + 71T^{2} \) |
| 73 | \( 1 + 3.67T + 73T^{2} \) |
| 79 | \( 1 - 12.0T + 79T^{2} \) |
| 83 | \( 1 - 6.96T + 83T^{2} \) |
| 89 | \( 1 - 14.7T + 89T^{2} \) |
| 97 | \( 1 + 14.0T + 97T^{2} \) |
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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
−8.950063718854691141463250810126, −7.993804404061252811374016586931, −7.60871258701898381381451743365, −6.51938681948832228338103213950, −5.54719576855081914646078530645, −5.14032693069240703980048037866, −3.41758258594709815051907820283, −2.58766360972516234988698839381, −1.79127796023450213402777558406, 0,
1.79127796023450213402777558406, 2.58766360972516234988698839381, 3.41758258594709815051907820283, 5.14032693069240703980048037866, 5.54719576855081914646078530645, 6.51938681948832228338103213950, 7.60871258701898381381451743365, 7.993804404061252811374016586931, 8.950063718854691141463250810126
