| L(s) = 1 | − 1.41·2-s − 1.41·3-s − 1.58·5-s + 2.00·6-s + 2.82·8-s − 0.999·9-s + 2.24·10-s + 4.24·11-s + 13-s + 2.24·15-s − 4.00·16-s − 1.41·17-s + 1.41·18-s + 7.24·19-s − 6·22-s − 5.82·23-s − 4·24-s − 2.48·25-s − 1.41·26-s + 5.65·27-s + 0.171·29-s − 3.17·30-s − 3.24·31-s − 6·33-s + 2.00·34-s + ⋯ |
| L(s) = 1 | − 1.00·2-s − 0.816·3-s − 0.709·5-s + 0.816·6-s + 0.999·8-s − 0.333·9-s + 0.709·10-s + 1.27·11-s + 0.277·13-s + 0.579·15-s − 1.00·16-s − 0.342·17-s + 0.333·18-s + 1.66·19-s − 1.27·22-s − 1.21·23-s − 0.816·24-s − 0.497·25-s − 0.277·26-s + 1.08·27-s + 0.0318·29-s − 0.579·30-s − 0.582·31-s − 1.04·33-s + 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + 1.41T + 2T^{2} \) |
| 3 | \( 1 + 1.41T + 3T^{2} \) |
| 5 | \( 1 + 1.58T + 5T^{2} \) |
| 11 | \( 1 - 4.24T + 11T^{2} \) |
| 17 | \( 1 + 1.41T + 17T^{2} \) |
| 19 | \( 1 - 7.24T + 19T^{2} \) |
| 23 | \( 1 + 5.82T + 23T^{2} \) |
| 29 | \( 1 - 0.171T + 29T^{2} \) |
| 31 | \( 1 + 3.24T + 31T^{2} \) |
| 37 | \( 1 - 2.24T + 37T^{2} \) |
| 41 | \( 1 + 8.82T + 41T^{2} \) |
| 43 | \( 1 + 5T + 43T^{2} \) |
| 47 | \( 1 + 1.58T + 47T^{2} \) |
| 53 | \( 1 + 0.171T + 53T^{2} \) |
| 59 | \( 1 + 0.343T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 + 14.4T + 67T^{2} \) |
| 71 | \( 1 + 13.0T + 71T^{2} \) |
| 73 | \( 1 - 9.24T + 73T^{2} \) |
| 79 | \( 1 - 15.4T + 79T^{2} \) |
| 83 | \( 1 + 13.2T + 83T^{2} \) |
| 89 | \( 1 + 1.58T + 89T^{2} \) |
| 97 | \( 1 + 11.7T + 97T^{2} \) |
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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
−10.04931387526865206844018821599, −9.289722327219775356453548631834, −8.465930869040071870541867222384, −7.65772624262341235406399982813, −6.72476143143247053380808563694, −5.68433579889334873119869343643, −4.55245601213709999230526775232, −3.53377262579408259094685718725, −1.42033141136442452973406263031, 0,
1.42033141136442452973406263031, 3.53377262579408259094685718725, 4.55245601213709999230526775232, 5.68433579889334873119869343643, 6.72476143143247053380808563694, 7.65772624262341235406399982813, 8.465930869040071870541867222384, 9.289722327219775356453548631834, 10.04931387526865206844018821599
