| L(s) = 1 | − 0.301·2-s − 1.85·3-s − 1.90·4-s + 0.558·6-s + 1.94·7-s + 1.17·8-s + 0.436·9-s − 11-s + 3.53·12-s − 3.75·13-s − 0.586·14-s + 3.46·16-s − 2.16·17-s − 0.131·18-s + 19-s − 3.60·21-s + 0.301·22-s + 4.03·23-s − 2.18·24-s + 1.13·26-s + 4.75·27-s − 3.71·28-s − 2.18·29-s + 3.21·31-s − 3.39·32-s + 1.85·33-s + 0.650·34-s + ⋯ |
| L(s) = 1 | − 0.212·2-s − 1.07·3-s − 0.954·4-s + 0.227·6-s + 0.735·7-s + 0.416·8-s + 0.145·9-s − 0.301·11-s + 1.02·12-s − 1.04·13-s − 0.156·14-s + 0.866·16-s − 0.524·17-s − 0.0309·18-s + 0.229·19-s − 0.787·21-s + 0.0642·22-s + 0.841·23-s − 0.445·24-s + 0.221·26-s + 0.914·27-s − 0.702·28-s − 0.406·29-s + 0.577·31-s − 0.600·32-s + 0.322·33-s + 0.111·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 0.301T + 2T^{2} \) |
| 3 | \( 1 + 1.85T + 3T^{2} \) |
| 7 | \( 1 - 1.94T + 7T^{2} \) |
| 13 | \( 1 + 3.75T + 13T^{2} \) |
| 17 | \( 1 + 2.16T + 17T^{2} \) |
| 23 | \( 1 - 4.03T + 23T^{2} \) |
| 29 | \( 1 + 2.18T + 29T^{2} \) |
| 31 | \( 1 - 3.21T + 31T^{2} \) |
| 37 | \( 1 + 5.64T + 37T^{2} \) |
| 41 | \( 1 + 3.52T + 41T^{2} \) |
| 43 | \( 1 - 2.74T + 43T^{2} \) |
| 47 | \( 1 + 1.87T + 47T^{2} \) |
| 53 | \( 1 - 4.59T + 53T^{2} \) |
| 59 | \( 1 - 3.10T + 59T^{2} \) |
| 61 | \( 1 + 10.0T + 61T^{2} \) |
| 67 | \( 1 - 12.0T + 67T^{2} \) |
| 71 | \( 1 - 6.42T + 71T^{2} \) |
| 73 | \( 1 - 10.7T + 73T^{2} \) |
| 79 | \( 1 - 10.8T + 79T^{2} \) |
| 83 | \( 1 - 8.12T + 83T^{2} \) |
| 89 | \( 1 + 11.1T + 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
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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
−7.915907907840985616297524161004, −7.14150134434057288199909460303, −6.37478776842590135329828098762, −5.30201762209013691513913639261, −5.09781326215345075882807627919, −4.48303363459717546353833818761, −3.40712955044850841210937146832, −2.21727363234810312449443273785, −0.979111868330556181048235223918, 0,
0.979111868330556181048235223918, 2.21727363234810312449443273785, 3.40712955044850841210937146832, 4.48303363459717546353833818761, 5.09781326215345075882807627919, 5.30201762209013691513913639261, 6.37478776842590135329828098762, 7.14150134434057288199909460303, 7.915907907840985616297524161004
