| L(s) = 1 | − 2.77·2-s + 2.60·3-s + 5.69·4-s − 7.23·6-s + 1.40·7-s − 10.2·8-s + 3.80·9-s − 4.51·11-s + 14.8·12-s + 3.66·13-s − 3.89·14-s + 17.0·16-s + 1.12·17-s − 10.5·18-s − 7.75·19-s + 3.66·21-s + 12.5·22-s − 8.71·23-s − 26.7·24-s − 10.1·26-s + 2.10·27-s + 8.00·28-s + 5.79·29-s + 10.1·31-s − 26.8·32-s − 11.7·33-s − 3.11·34-s + ⋯ |
| L(s) = 1 | − 1.96·2-s + 1.50·3-s + 2.84·4-s − 2.95·6-s + 0.530·7-s − 3.62·8-s + 1.26·9-s − 1.36·11-s + 4.29·12-s + 1.01·13-s − 1.04·14-s + 4.26·16-s + 0.272·17-s − 2.48·18-s − 1.77·19-s + 0.799·21-s + 2.66·22-s − 1.81·23-s − 5.46·24-s − 1.99·26-s + 0.405·27-s + 1.51·28-s + 1.07·29-s + 1.81·31-s − 4.74·32-s − 2.04·33-s − 0.533·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 241 | \( 1 + T \) |
| good | 2 | \( 1 + 2.77T + 2T^{2} \) |
| 3 | \( 1 - 2.60T + 3T^{2} \) |
| 7 | \( 1 - 1.40T + 7T^{2} \) |
| 11 | \( 1 + 4.51T + 11T^{2} \) |
| 13 | \( 1 - 3.66T + 13T^{2} \) |
| 17 | \( 1 - 1.12T + 17T^{2} \) |
| 19 | \( 1 + 7.75T + 19T^{2} \) |
| 23 | \( 1 + 8.71T + 23T^{2} \) |
| 29 | \( 1 - 5.79T + 29T^{2} \) |
| 31 | \( 1 - 10.1T + 31T^{2} \) |
| 37 | \( 1 - 2.30T + 37T^{2} \) |
| 41 | \( 1 + 1.43T + 41T^{2} \) |
| 43 | \( 1 + 9.39T + 43T^{2} \) |
| 47 | \( 1 + 11.9T + 47T^{2} \) |
| 53 | \( 1 + 9.40T + 53T^{2} \) |
| 59 | \( 1 - 3.44T + 59T^{2} \) |
| 61 | \( 1 - 5.18T + 61T^{2} \) |
| 67 | \( 1 + 11.1T + 67T^{2} \) |
| 71 | \( 1 + 1.58T + 71T^{2} \) |
| 73 | \( 1 + 7.27T + 73T^{2} \) |
| 79 | \( 1 - 8.45T + 79T^{2} \) |
| 83 | \( 1 - 15.5T + 83T^{2} \) |
| 89 | \( 1 + 4.71T + 89T^{2} \) |
| 97 | \( 1 + 2.12T + 97T^{2} \) |
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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
−8.150114333689876149367892498986, −7.66608552431820965169122042195, −6.48145586630348990453408457326, −6.19124873000770779295117719080, −4.74179248356899137642476109329, −3.54527983077790722928132282744, −2.76549889757394245191587912068, −2.13143997235517274423667534837, −1.46030948893890875434246553076, 0,
1.46030948893890875434246553076, 2.13143997235517274423667534837, 2.76549889757394245191587912068, 3.54527983077790722928132282744, 4.74179248356899137642476109329, 6.19124873000770779295117719080, 6.48145586630348990453408457326, 7.66608552431820965169122042195, 8.150114333689876149367892498986
