| L(s) = 1 | + 2.53·2-s − 3-s + 4.43·4-s − 2.53·6-s + 1.04·7-s + 6.19·8-s + 9-s + 2.97·11-s − 4.43·12-s + 5.66·13-s + 2.64·14-s + 6.83·16-s − 5.08·17-s + 2.53·18-s − 5.37·19-s − 1.04·21-s + 7.53·22-s + 3.86·23-s − 6.19·24-s + 14.3·26-s − 27-s + 4.61·28-s + 0.679·29-s + 0.850·31-s + 4.95·32-s − 2.97·33-s − 12.9·34-s + ⋯ |
| L(s) = 1 | + 1.79·2-s − 0.577·3-s + 2.21·4-s − 1.03·6-s + 0.393·7-s + 2.18·8-s + 0.333·9-s + 0.895·11-s − 1.28·12-s + 1.57·13-s + 0.705·14-s + 1.70·16-s − 1.23·17-s + 0.598·18-s − 1.23·19-s − 0.227·21-s + 1.60·22-s + 0.804·23-s − 1.26·24-s + 2.82·26-s − 0.192·27-s + 0.873·28-s + 0.126·29-s + 0.152·31-s + 0.875·32-s − 0.517·33-s − 2.21·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.963194540\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.963194540\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 2 | \( 1 - 2.53T + 2T^{2} \) |
| 7 | \( 1 - 1.04T + 7T^{2} \) |
| 11 | \( 1 - 2.97T + 11T^{2} \) |
| 13 | \( 1 - 5.66T + 13T^{2} \) |
| 17 | \( 1 + 5.08T + 17T^{2} \) |
| 19 | \( 1 + 5.37T + 19T^{2} \) |
| 23 | \( 1 - 3.86T + 23T^{2} \) |
| 29 | \( 1 - 0.679T + 29T^{2} \) |
| 31 | \( 1 - 0.850T + 31T^{2} \) |
| 37 | \( 1 + 1.61T + 37T^{2} \) |
| 41 | \( 1 - 1.16T + 41T^{2} \) |
| 43 | \( 1 - 5.68T + 43T^{2} \) |
| 47 | \( 1 - 3.28T + 47T^{2} \) |
| 53 | \( 1 - 12.6T + 53T^{2} \) |
| 59 | \( 1 - 3.21T + 59T^{2} \) |
| 61 | \( 1 + 5.42T + 61T^{2} \) |
| 67 | \( 1 + 0.929T + 67T^{2} \) |
| 71 | \( 1 + 1.41T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 + 1.44T + 79T^{2} \) |
| 83 | \( 1 + 11.4T + 83T^{2} \) |
| 89 | \( 1 - 9.07T + 89T^{2} \) |
| 97 | \( 1 + 6.02T + 97T^{2} \) |
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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
−9.118462917690198840513046517793, −8.398950665637307366353301471005, −7.09727066814418356724825728384, −6.48908366108361147448701299742, −6.00916313858621882675854258873, −5.08750251736620216558352570923, −4.21249895267563007605500631865, −3.82685654000343620602718516805, −2.50003513441518027904301912754, −1.38869621811313374227001036047,
1.38869621811313374227001036047, 2.50003513441518027904301912754, 3.82685654000343620602718516805, 4.21249895267563007605500631865, 5.08750251736620216558352570923, 6.00916313858621882675854258873, 6.48908366108361147448701299742, 7.09727066814418356724825728384, 8.398950665637307366353301471005, 9.118462917690198840513046517793
