| L(s) = 1 | + 1.09·2-s − 1.55·3-s − 0.804·4-s + 0.717·5-s − 1.69·6-s + 5.03·7-s − 3.06·8-s − 0.587·9-s + 0.784·10-s + 5.82·11-s + 1.25·12-s + 5.50·14-s − 1.11·15-s − 1.74·16-s − 1.01·17-s − 0.642·18-s − 7.37·19-s − 0.577·20-s − 7.81·21-s + 6.37·22-s − 7.58·23-s + 4.76·24-s − 4.48·25-s + 5.57·27-s − 4.05·28-s − 5.08·29-s − 1.21·30-s + ⋯ |
| L(s) = 1 | + 0.773·2-s − 0.896·3-s − 0.402·4-s + 0.320·5-s − 0.693·6-s + 1.90·7-s − 1.08·8-s − 0.195·9-s + 0.248·10-s + 1.75·11-s + 0.360·12-s + 1.47·14-s − 0.287·15-s − 0.435·16-s − 0.246·17-s − 0.151·18-s − 1.69·19-s − 0.129·20-s − 1.70·21-s + 1.35·22-s − 1.58·23-s + 0.972·24-s − 0.896·25-s + 1.07·27-s − 0.765·28-s − 0.944·29-s − 0.222·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 - 1.09T + 2T^{2} \) |
| 3 | \( 1 + 1.55T + 3T^{2} \) |
| 5 | \( 1 - 0.717T + 5T^{2} \) |
| 7 | \( 1 - 5.03T + 7T^{2} \) |
| 11 | \( 1 - 5.82T + 11T^{2} \) |
| 17 | \( 1 + 1.01T + 17T^{2} \) |
| 19 | \( 1 + 7.37T + 19T^{2} \) |
| 23 | \( 1 + 7.58T + 23T^{2} \) |
| 29 | \( 1 + 5.08T + 29T^{2} \) |
| 37 | \( 1 + 3.23T + 37T^{2} \) |
| 41 | \( 1 - 1.25T + 41T^{2} \) |
| 43 | \( 1 - 3.95T + 43T^{2} \) |
| 47 | \( 1 + 9.68T + 47T^{2} \) |
| 53 | \( 1 + 7.60T + 53T^{2} \) |
| 59 | \( 1 - 4.76T + 59T^{2} \) |
| 61 | \( 1 - 6.02T + 61T^{2} \) |
| 67 | \( 1 + 6.83T + 67T^{2} \) |
| 71 | \( 1 + 3.60T + 71T^{2} \) |
| 73 | \( 1 - 4.60T + 73T^{2} \) |
| 79 | \( 1 - 0.810T + 79T^{2} \) |
| 83 | \( 1 + 11.6T + 83T^{2} \) |
| 89 | \( 1 + 0.574T + 89T^{2} \) |
| 97 | \( 1 - 4.21T + 97T^{2} \) |
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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
−7.983923393548552662793916219258, −6.78861877837902801935922918119, −6.07448644548355755866613319938, −5.69306885460426841681123078717, −4.85589114996088724098243044443, −4.25329705560820672979439255661, −3.80617311652461034061870175298, −2.19470872588037543490250474313, −1.47198651089890403850641208198, 0,
1.47198651089890403850641208198, 2.19470872588037543490250474313, 3.80617311652461034061870175298, 4.25329705560820672979439255661, 4.85589114996088724098243044443, 5.69306885460426841681123078717, 6.07448644548355755866613319938, 6.78861877837902801935922918119, 7.983923393548552662793916219258
