| L(s) = 1 | − 3.70·5-s + 7-s + 4.57·11-s − 5.23·13-s + 0.874·17-s − 19-s + 1.08·23-s + 8.70·25-s + 4.78·29-s − 7.23·31-s − 3.70·35-s − 6·37-s + 4.57·41-s + 12.1·43-s − 10.0·47-s + 49-s + 9.35·53-s − 16.9·55-s + 12.6·59-s − 10.9·61-s + 19.3·65-s + 6.47·67-s − 0.206·71-s − 12.4·73-s + 4.57·77-s − 12.9·79-s − 16.7·83-s + ⋯ |
| L(s) = 1 | − 1.65·5-s + 0.377·7-s + 1.37·11-s − 1.45·13-s + 0.211·17-s − 0.229·19-s + 0.225·23-s + 1.74·25-s + 0.888·29-s − 1.29·31-s − 0.625·35-s − 0.986·37-s + 0.714·41-s + 1.85·43-s − 1.46·47-s + 0.142·49-s + 1.28·53-s − 2.28·55-s + 1.64·59-s − 1.40·61-s + 2.40·65-s + 0.790·67-s − 0.0244·71-s − 1.45·73-s + 0.521·77-s − 1.45·79-s − 1.84·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + 3.70T + 5T^{2} \) |
| 11 | \( 1 - 4.57T + 11T^{2} \) |
| 13 | \( 1 + 5.23T + 13T^{2} \) |
| 17 | \( 1 - 0.874T + 17T^{2} \) |
| 23 | \( 1 - 1.08T + 23T^{2} \) |
| 29 | \( 1 - 4.78T + 29T^{2} \) |
| 31 | \( 1 + 7.23T + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 - 4.57T + 41T^{2} \) |
| 43 | \( 1 - 12.1T + 43T^{2} \) |
| 47 | \( 1 + 10.0T + 47T^{2} \) |
| 53 | \( 1 - 9.35T + 53T^{2} \) |
| 59 | \( 1 - 12.6T + 59T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 - 6.47T + 67T^{2} \) |
| 71 | \( 1 + 0.206T + 71T^{2} \) |
| 73 | \( 1 + 12.4T + 73T^{2} \) |
| 79 | \( 1 + 12.9T + 79T^{2} \) |
| 83 | \( 1 + 16.7T + 83T^{2} \) |
| 89 | \( 1 - 4.57T + 89T^{2} \) |
| 97 | \( 1 - 9.41T + 97T^{2} \) |
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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
−7.79809218198884431436065476620, −7.25073635101609668919166740786, −6.80320011338635448717674888719, −5.65456999533785586061157805079, −4.72067390786763167319210219407, −4.17851361570849248701947861362, −3.51801928325114189344954398220, −2.50861041805258019857604858802, −1.20905818622487886292865292554, 0,
1.20905818622487886292865292554, 2.50861041805258019857604858802, 3.51801928325114189344954398220, 4.17851361570849248701947861362, 4.72067390786763167319210219407, 5.65456999533785586061157805079, 6.80320011338635448717674888719, 7.25073635101609668919166740786, 7.79809218198884431436065476620
