| L(s) = 1 | − 2.18·2-s + 1.79·3-s + 2.79·4-s + 2.18·5-s − 3.92·6-s − 1.73·8-s + 0.208·9-s − 4.79·10-s − 1.27·11-s + 4.99·12-s + 3.92·15-s − 1.79·16-s + 3·17-s − 0.456·18-s + 6.56·19-s + 6.10·20-s + 2.79·22-s − 7.58·23-s − 3.10·24-s − 0.208·25-s − 5.00·27-s − 2.20·29-s − 8.58·30-s − 8.66·31-s + 7.38·32-s − 2.28·33-s − 6.56·34-s + ⋯ |
| L(s) = 1 | − 1.54·2-s + 1.03·3-s + 1.39·4-s + 0.978·5-s − 1.60·6-s − 0.612·8-s + 0.0695·9-s − 1.51·10-s − 0.384·11-s + 1.44·12-s + 1.01·15-s − 0.447·16-s + 0.727·17-s − 0.107·18-s + 1.50·19-s + 1.36·20-s + 0.595·22-s − 1.58·23-s − 0.633·24-s − 0.0417·25-s − 0.962·27-s − 0.410·29-s − 1.56·30-s − 1.55·31-s + 1.30·32-s − 0.397·33-s − 1.12·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.18T + 2T^{2} \) |
| 3 | \( 1 - 1.79T + 3T^{2} \) |
| 5 | \( 1 - 2.18T + 5T^{2} \) |
| 11 | \( 1 + 1.27T + 11T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 19 | \( 1 - 6.56T + 19T^{2} \) |
| 23 | \( 1 + 7.58T + 23T^{2} \) |
| 29 | \( 1 + 2.20T + 29T^{2} \) |
| 31 | \( 1 + 8.66T + 31T^{2} \) |
| 37 | \( 1 - 6.92T + 37T^{2} \) |
| 41 | \( 1 - 2.55T + 41T^{2} \) |
| 43 | \( 1 - 4.37T + 43T^{2} \) |
| 47 | \( 1 + 4.28T + 47T^{2} \) |
| 53 | \( 1 + 12.1T + 53T^{2} \) |
| 59 | \( 1 + 8.85T + 59T^{2} \) |
| 61 | \( 1 + 12.7T + 61T^{2} \) |
| 67 | \( 1 + 11.4T + 67T^{2} \) |
| 71 | \( 1 + 0.913T + 71T^{2} \) |
| 73 | \( 1 + 3.46T + 73T^{2} \) |
| 79 | \( 1 + 6T + 79T^{2} \) |
| 83 | \( 1 + 3.55T + 83T^{2} \) |
| 89 | \( 1 - 2.91T + 89T^{2} \) |
| 97 | \( 1 - 15.2T + 97T^{2} \) |
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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
−7.69475275815019337060702821076, −7.34919654796750472633502706421, −6.00319675643851011599563516042, −5.82283965533704249864137433334, −4.63917240028289197956813971939, −3.48048848693804697474930233579, −2.78690730538629641564743910182, −1.95129562410777885641795288495, −1.42370763263260325839201145816, 0,
1.42370763263260325839201145816, 1.95129562410777885641795288495, 2.78690730538629641564743910182, 3.48048848693804697474930233579, 4.63917240028289197956813971939, 5.82283965533704249864137433334, 6.00319675643851011599563516042, 7.34919654796750472633502706421, 7.69475275815019337060702821076
