| L(s) = 1 | + 2.24·2-s + 1.20·3-s + 3.05·4-s − 0.269·5-s + 2.70·6-s − 0.523·7-s + 2.36·8-s − 1.55·9-s − 0.605·10-s + 5.54·11-s + 3.67·12-s + 3.85·13-s − 1.17·14-s − 0.324·15-s − 0.785·16-s − 1.23·17-s − 3.48·18-s + 1.15·19-s − 0.822·20-s − 0.630·21-s + 12.4·22-s − 23-s + 2.85·24-s − 4.92·25-s + 8.66·26-s − 5.47·27-s − 1.59·28-s + ⋯ |
| L(s) = 1 | + 1.58·2-s + 0.695·3-s + 1.52·4-s − 0.120·5-s + 1.10·6-s − 0.197·7-s + 0.836·8-s − 0.516·9-s − 0.191·10-s + 1.67·11-s + 1.06·12-s + 1.06·13-s − 0.314·14-s − 0.0837·15-s − 0.196·16-s − 0.298·17-s − 0.821·18-s + 0.265·19-s − 0.183·20-s − 0.137·21-s + 2.65·22-s − 0.208·23-s + 0.581·24-s − 0.985·25-s + 1.69·26-s − 1.05·27-s − 0.302·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.216585304\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.216585304\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 2.24T + 2T^{2} \) |
| 3 | \( 1 - 1.20T + 3T^{2} \) |
| 5 | \( 1 + 0.269T + 5T^{2} \) |
| 7 | \( 1 + 0.523T + 7T^{2} \) |
| 11 | \( 1 - 5.54T + 11T^{2} \) |
| 13 | \( 1 - 3.85T + 13T^{2} \) |
| 17 | \( 1 + 1.23T + 17T^{2} \) |
| 19 | \( 1 - 1.15T + 19T^{2} \) |
| 31 | \( 1 - 1.02T + 31T^{2} \) |
| 37 | \( 1 + 3.22T + 37T^{2} \) |
| 41 | \( 1 + 3.67T + 41T^{2} \) |
| 43 | \( 1 - 2.77T + 43T^{2} \) |
| 47 | \( 1 + 6.71T + 47T^{2} \) |
| 53 | \( 1 + 2.58T + 53T^{2} \) |
| 59 | \( 1 + 9.36T + 59T^{2} \) |
| 61 | \( 1 + 12.0T + 61T^{2} \) |
| 67 | \( 1 + 2.14T + 67T^{2} \) |
| 71 | \( 1 - 4.70T + 71T^{2} \) |
| 73 | \( 1 - 11.8T + 73T^{2} \) |
| 79 | \( 1 - 15.5T + 79T^{2} \) |
| 83 | \( 1 - 8.53T + 83T^{2} \) |
| 89 | \( 1 - 12.2T + 89T^{2} \) |
| 97 | \( 1 - 3.00T + 97T^{2} \) |
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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
−10.93969506333758587327689703395, −9.467938589194665679551139797206, −8.838722084440643321788146529132, −7.78942627139945203482996620934, −6.44812521966842676925799596578, −6.13503792391611574029238424995, −4.82444608758861004781972093892, −3.71855012230735758117266263747, −3.34387394586028050846093217469, −1.88522951772703649404046500246,
1.88522951772703649404046500246, 3.34387394586028050846093217469, 3.71855012230735758117266263747, 4.82444608758861004781972093892, 6.13503792391611574029238424995, 6.44812521966842676925799596578, 7.78942627139945203482996620934, 8.838722084440643321788146529132, 9.467938589194665679551139797206, 10.93969506333758587327689703395
