| L(s) = 1 | − 0.347·2-s − 3-s − 1.87·4-s − 5-s + 0.347·6-s + 1.34·8-s + 9-s + 0.347·10-s + 11-s + 1.87·12-s − 5.01·13-s + 15-s + 3.28·16-s − 6.74·17-s − 0.347·18-s − 6.66·19-s + 1.87·20-s − 0.347·22-s − 2.01·23-s − 1.34·24-s + 25-s + 1.74·26-s − 27-s − 3.99·29-s − 0.347·30-s − 7.39·31-s − 3.84·32-s + ⋯ |
| L(s) = 1 | − 0.245·2-s − 0.577·3-s − 0.939·4-s − 0.447·5-s + 0.141·6-s + 0.476·8-s + 0.333·9-s + 0.109·10-s + 0.301·11-s + 0.542·12-s − 1.39·13-s + 0.258·15-s + 0.822·16-s − 1.63·17-s − 0.0819·18-s − 1.52·19-s + 0.420·20-s − 0.0741·22-s − 0.421·23-s − 0.275·24-s + 0.200·25-s + 0.341·26-s − 0.192·27-s − 0.741·29-s − 0.0634·30-s − 1.32·31-s − 0.678·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8085 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8085 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.04026982565\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04026982565\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + 0.347T + 2T^{2} \) |
| 13 | \( 1 + 5.01T + 13T^{2} \) |
| 17 | \( 1 + 6.74T + 17T^{2} \) |
| 19 | \( 1 + 6.66T + 19T^{2} \) |
| 23 | \( 1 + 2.01T + 23T^{2} \) |
| 29 | \( 1 + 3.99T + 29T^{2} \) |
| 31 | \( 1 + 7.39T + 31T^{2} \) |
| 37 | \( 1 + 3.29T + 37T^{2} \) |
| 41 | \( 1 - 6.77T + 41T^{2} \) |
| 43 | \( 1 - 10.4T + 43T^{2} \) |
| 47 | \( 1 + 3.28T + 47T^{2} \) |
| 53 | \( 1 - 0.122T + 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 + 2.63T + 61T^{2} \) |
| 67 | \( 1 + 2.00T + 67T^{2} \) |
| 71 | \( 1 + 15.7T + 71T^{2} \) |
| 73 | \( 1 + 12.1T + 73T^{2} \) |
| 79 | \( 1 - 2.65T + 79T^{2} \) |
| 83 | \( 1 - 1.18T + 83T^{2} \) |
| 89 | \( 1 + 4.23T + 89T^{2} \) |
| 97 | \( 1 - 4.83T + 97T^{2} \) |
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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
−7.63879340303196243916746221091, −7.35574877727689215847477647976, −6.41866691505769632076985715558, −5.74429250063042493182452604956, −4.82845951753117615274566359890, −4.36479121065938507518076316299, −3.87063731042437082187458512427, −2.55228188702362483833137486965, −1.64550763823580437384241101477, −0.10482318073637860170616898854,
0.10482318073637860170616898854, 1.64550763823580437384241101477, 2.55228188702362483833137486965, 3.87063731042437082187458512427, 4.36479121065938507518076316299, 4.82845951753117615274566359890, 5.74429250063042493182452604956, 6.41866691505769632076985715558, 7.35574877727689215847477647976, 7.63879340303196243916746221091
