| L(s) = 1 | − 2.57·2-s + 4.64·4-s + 4.69·7-s − 6.81·8-s + 3.11·11-s − 5.07·13-s − 12.1·14-s + 8.28·16-s − 1.40·17-s − 3.76·19-s − 8.03·22-s − 5.71·23-s + 13.0·26-s + 21.8·28-s − 29-s − 2.23·31-s − 7.73·32-s + 3.60·34-s + 5.79·37-s + 9.70·38-s + 10.6·41-s − 8.89·43-s + 14.4·44-s + 14.7·46-s − 3.62·47-s + 15.0·49-s − 23.5·52-s + ⋯ |
| L(s) = 1 | − 1.82·2-s + 2.32·4-s + 1.77·7-s − 2.41·8-s + 0.939·11-s − 1.40·13-s − 3.23·14-s + 2.07·16-s − 0.339·17-s − 0.863·19-s − 1.71·22-s − 1.19·23-s + 2.56·26-s + 4.12·28-s − 0.185·29-s − 0.400·31-s − 1.36·32-s + 0.619·34-s + 0.952·37-s + 1.57·38-s + 1.67·41-s − 1.35·43-s + 2.18·44-s + 2.17·46-s − 0.529·47-s + 2.15·49-s − 3.27·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 2.57T + 2T^{2} \) |
| 7 | \( 1 - 4.69T + 7T^{2} \) |
| 11 | \( 1 - 3.11T + 11T^{2} \) |
| 13 | \( 1 + 5.07T + 13T^{2} \) |
| 17 | \( 1 + 1.40T + 17T^{2} \) |
| 19 | \( 1 + 3.76T + 19T^{2} \) |
| 23 | \( 1 + 5.71T + 23T^{2} \) |
| 31 | \( 1 + 2.23T + 31T^{2} \) |
| 37 | \( 1 - 5.79T + 37T^{2} \) |
| 41 | \( 1 - 10.6T + 41T^{2} \) |
| 43 | \( 1 + 8.89T + 43T^{2} \) |
| 47 | \( 1 + 3.62T + 47T^{2} \) |
| 53 | \( 1 - 0.948T + 53T^{2} \) |
| 59 | \( 1 - 8.53T + 59T^{2} \) |
| 61 | \( 1 + 6.21T + 61T^{2} \) |
| 67 | \( 1 + 13.9T + 67T^{2} \) |
| 71 | \( 1 + 5.88T + 71T^{2} \) |
| 73 | \( 1 - 7.08T + 73T^{2} \) |
| 79 | \( 1 - 7.31T + 79T^{2} \) |
| 83 | \( 1 + 13.6T + 83T^{2} \) |
| 89 | \( 1 - 7.98T + 89T^{2} \) |
| 97 | \( 1 + 13.8T + 97T^{2} \) |
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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
−7.78280190628978762509429401210, −7.35485435371872029782794761762, −6.56803133553546558619632274712, −5.78762875277821434528167501092, −4.74665820987326923863898530779, −4.08683926466824348439100413826, −2.55318381383418278076963569558, −1.96613109548948213049500434777, −1.27220129901495762124123306942, 0,
1.27220129901495762124123306942, 1.96613109548948213049500434777, 2.55318381383418278076963569558, 4.08683926466824348439100413826, 4.74665820987326923863898530779, 5.78762875277821434528167501092, 6.56803133553546558619632274712, 7.35485435371872029782794761762, 7.78280190628978762509429401210
