| L(s) = 1 | + 0.185·2-s + 3-s − 1.96·4-s + 0.185·6-s − 0.737·8-s + 9-s − 5.19·11-s − 1.96·12-s − 2.77·13-s + 3.79·16-s + 2.56·17-s + 0.185·18-s − 5.08·19-s − 0.965·22-s − 2.09·23-s − 0.737·24-s − 0.516·26-s + 27-s + 4.66·29-s − 3.49·31-s + 2.17·32-s − 5.19·33-s + 0.476·34-s − 1.96·36-s + 6.20·37-s − 0.944·38-s − 2.77·39-s + ⋯ |
| L(s) = 1 | + 0.131·2-s + 0.577·3-s − 0.982·4-s + 0.0758·6-s − 0.260·8-s + 0.333·9-s − 1.56·11-s − 0.567·12-s − 0.770·13-s + 0.948·16-s + 0.622·17-s + 0.0438·18-s − 1.16·19-s − 0.205·22-s − 0.436·23-s − 0.150·24-s − 0.101·26-s + 0.192·27-s + 0.866·29-s − 0.628·31-s + 0.385·32-s − 0.904·33-s + 0.0817·34-s − 0.327·36-s + 1.01·37-s − 0.153·38-s − 0.445·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.391832996\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.391832996\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 0.185T + 2T^{2} \) |
| 11 | \( 1 + 5.19T + 11T^{2} \) |
| 13 | \( 1 + 2.77T + 13T^{2} \) |
| 17 | \( 1 - 2.56T + 17T^{2} \) |
| 19 | \( 1 + 5.08T + 19T^{2} \) |
| 23 | \( 1 + 2.09T + 23T^{2} \) |
| 29 | \( 1 - 4.66T + 29T^{2} \) |
| 31 | \( 1 + 3.49T + 31T^{2} \) |
| 37 | \( 1 - 6.20T + 37T^{2} \) |
| 41 | \( 1 - 7.47T + 41T^{2} \) |
| 43 | \( 1 - 10.4T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 + 4.15T + 53T^{2} \) |
| 59 | \( 1 + 0.997T + 59T^{2} \) |
| 61 | \( 1 - 10.8T + 61T^{2} \) |
| 67 | \( 1 - 5.94T + 67T^{2} \) |
| 71 | \( 1 + 11.7T + 71T^{2} \) |
| 73 | \( 1 + 3.97T + 73T^{2} \) |
| 79 | \( 1 - 3.23T + 79T^{2} \) |
| 83 | \( 1 + 16.7T + 83T^{2} \) |
| 89 | \( 1 - 4.68T + 89T^{2} \) |
| 97 | \( 1 - 14.4T + 97T^{2} \) |
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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
−8.571702538068054724516578426310, −7.74582789757553139236590181786, −7.47254799785700971416368874312, −6.10020466859196674984230664574, −5.45076611367204720915121020877, −4.60126474367195640489828063718, −4.04366113045509980520665971127, −2.91511898301884970319532335010, −2.27915860417352155759525098868, −0.63870405430360973043362896765,
0.63870405430360973043362896765, 2.27915860417352155759525098868, 2.91511898301884970319532335010, 4.04366113045509980520665971127, 4.60126474367195640489828063718, 5.45076611367204720915121020877, 6.10020466859196674984230664574, 7.47254799785700971416368874312, 7.74582789757553139236590181786, 8.571702538068054724516578426310
