| L(s) = 1 | + 2.60·2-s + 4.81·4-s − 1.35·7-s + 7.33·8-s − 2.89·11-s + 4.64·13-s − 3.52·14-s + 9.53·16-s + 6.57·17-s − 4.94·19-s − 7.56·22-s − 3.70·23-s + 12.1·26-s − 6.50·28-s + 2.99·29-s + 31-s + 10.1·32-s + 17.1·34-s + 9.62·37-s − 12.9·38-s + 7.00·41-s + 2.99·43-s − 13.9·44-s − 9.66·46-s + 7.98·47-s − 5.17·49-s + 22.3·52-s + ⋯ |
| L(s) = 1 | + 1.84·2-s + 2.40·4-s − 0.510·7-s + 2.59·8-s − 0.873·11-s + 1.28·13-s − 0.942·14-s + 2.38·16-s + 1.59·17-s − 1.13·19-s − 1.61·22-s − 0.772·23-s + 2.37·26-s − 1.22·28-s + 0.555·29-s + 0.179·31-s + 1.80·32-s + 2.94·34-s + 1.58·37-s − 2.09·38-s + 1.09·41-s + 0.457·43-s − 2.10·44-s − 1.42·46-s + 1.16·47-s − 0.738·49-s + 3.10·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.040822174\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.040822174\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 - 2.60T + 2T^{2} \) |
| 7 | \( 1 + 1.35T + 7T^{2} \) |
| 11 | \( 1 + 2.89T + 11T^{2} \) |
| 13 | \( 1 - 4.64T + 13T^{2} \) |
| 17 | \( 1 - 6.57T + 17T^{2} \) |
| 19 | \( 1 + 4.94T + 19T^{2} \) |
| 23 | \( 1 + 3.70T + 23T^{2} \) |
| 29 | \( 1 - 2.99T + 29T^{2} \) |
| 37 | \( 1 - 9.62T + 37T^{2} \) |
| 41 | \( 1 - 7.00T + 41T^{2} \) |
| 43 | \( 1 - 2.99T + 43T^{2} \) |
| 47 | \( 1 - 7.98T + 47T^{2} \) |
| 53 | \( 1 - 13.0T + 53T^{2} \) |
| 59 | \( 1 + 4.50T + 59T^{2} \) |
| 61 | \( 1 - 7.87T + 61T^{2} \) |
| 67 | \( 1 - 4.37T + 67T^{2} \) |
| 71 | \( 1 - 11.0T + 71T^{2} \) |
| 73 | \( 1 + 4.81T + 73T^{2} \) |
| 79 | \( 1 - 5.87T + 79T^{2} \) |
| 83 | \( 1 + 0.458T + 83T^{2} \) |
| 89 | \( 1 + 8.57T + 89T^{2} \) |
| 97 | \( 1 - 0.397T + 97T^{2} \) |
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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
−7.83163989742575408185214169438, −6.92698626540438068110755204223, −6.19397125035568520498542076617, −5.80891443918109742220501805307, −5.19968111435645022098203884161, −4.12336972451236967386118925965, −3.85962700772992669772735792366, −2.87905244587889832671920004939, −2.35031930183474637957986382859, −1.06156066570043286455631853292,
1.06156066570043286455631853292, 2.35031930183474637957986382859, 2.87905244587889832671920004939, 3.85962700772992669772735792366, 4.12336972451236967386118925965, 5.19968111435645022098203884161, 5.80891443918109742220501805307, 6.19397125035568520498542076617, 6.92698626540438068110755204223, 7.83163989742575408185214169438
