| L(s) = 1 | − 2-s − 0.394·3-s + 4-s + 0.507·5-s + 0.394·6-s + 1.24·7-s − 8-s − 2.84·9-s − 0.507·10-s − 2.69·11-s − 0.394·12-s − 1.24·14-s − 0.200·15-s + 16-s + 4.56·17-s + 2.84·18-s + 19-s + 0.507·20-s − 0.493·21-s + 2.69·22-s + 2.24·23-s + 0.394·24-s − 4.74·25-s + 2.30·27-s + 1.24·28-s − 9.14·29-s + 0.200·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.227·3-s + 0.5·4-s + 0.227·5-s + 0.161·6-s + 0.472·7-s − 0.353·8-s − 0.948·9-s − 0.160·10-s − 0.813·11-s − 0.113·12-s − 0.333·14-s − 0.0517·15-s + 0.250·16-s + 1.10·17-s + 0.670·18-s + 0.229·19-s + 0.113·20-s − 0.107·21-s + 0.575·22-s + 0.469·23-s + 0.0805·24-s − 0.948·25-s + 0.443·27-s + 0.236·28-s − 1.69·29-s + 0.0365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 13 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 0.394T + 3T^{2} \) |
| 5 | \( 1 - 0.507T + 5T^{2} \) |
| 7 | \( 1 - 1.24T + 7T^{2} \) |
| 11 | \( 1 + 2.69T + 11T^{2} \) |
| 17 | \( 1 - 4.56T + 17T^{2} \) |
| 23 | \( 1 - 2.24T + 23T^{2} \) |
| 29 | \( 1 + 9.14T + 29T^{2} \) |
| 31 | \( 1 + 1.74T + 31T^{2} \) |
| 37 | \( 1 - 8.71T + 37T^{2} \) |
| 41 | \( 1 - 9.55T + 41T^{2} \) |
| 43 | \( 1 - 3.72T + 43T^{2} \) |
| 47 | \( 1 + 8.96T + 47T^{2} \) |
| 53 | \( 1 - 5.80T + 53T^{2} \) |
| 59 | \( 1 - 8.65T + 59T^{2} \) |
| 61 | \( 1 + 0.618T + 61T^{2} \) |
| 67 | \( 1 + 12.2T + 67T^{2} \) |
| 71 | \( 1 - 0.995T + 71T^{2} \) |
| 73 | \( 1 + 0.150T + 73T^{2} \) |
| 79 | \( 1 + 5.80T + 79T^{2} \) |
| 83 | \( 1 - 8.15T + 83T^{2} \) |
| 89 | \( 1 + 2.44T + 89T^{2} \) |
| 97 | \( 1 - 5.19T + 97T^{2} \) |
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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
−7.84563402028598462129759908310, −7.22708471438457794828914602616, −6.10026213962523413042353586108, −5.65964983515575906826101433825, −5.07883815644240578003104018287, −3.91414072381858765453583083238, −2.96486873984333480067799076797, −2.24539422758976895782411909884, −1.17885855013737209228523979543, 0,
1.17885855013737209228523979543, 2.24539422758976895782411909884, 2.96486873984333480067799076797, 3.91414072381858765453583083238, 5.07883815644240578003104018287, 5.65964983515575906826101433825, 6.10026213962523413042353586108, 7.22708471438457794828914602616, 7.84563402028598462129759908310
