| L(s) = 1 | − 2.75·2-s + 3-s + 5.56·4-s + 0.854·5-s − 2.75·6-s + 7-s − 9.80·8-s + 9-s − 2.34·10-s + 4.74·11-s + 5.56·12-s − 1.71·13-s − 2.75·14-s + 0.854·15-s + 15.8·16-s − 7.45·17-s − 2.75·18-s + 0.607·19-s + 4.75·20-s + 21-s − 13.0·22-s − 1.05·23-s − 9.80·24-s − 4.27·25-s + 4.72·26-s + 27-s + 5.56·28-s + ⋯ |
| L(s) = 1 | − 1.94·2-s + 0.577·3-s + 2.78·4-s + 0.381·5-s − 1.12·6-s + 0.377·7-s − 3.46·8-s + 0.333·9-s − 0.742·10-s + 1.43·11-s + 1.60·12-s − 0.476·13-s − 0.735·14-s + 0.220·15-s + 3.95·16-s − 1.80·17-s − 0.648·18-s + 0.139·19-s + 1.06·20-s + 0.218·21-s − 2.78·22-s − 0.219·23-s − 2.00·24-s − 0.854·25-s + 0.925·26-s + 0.192·27-s + 1.05·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 383 | \( 1 + T \) |
| good | 2 | \( 1 + 2.75T + 2T^{2} \) |
| 5 | \( 1 - 0.854T + 5T^{2} \) |
| 11 | \( 1 - 4.74T + 11T^{2} \) |
| 13 | \( 1 + 1.71T + 13T^{2} \) |
| 17 | \( 1 + 7.45T + 17T^{2} \) |
| 19 | \( 1 - 0.607T + 19T^{2} \) |
| 23 | \( 1 + 1.05T + 23T^{2} \) |
| 29 | \( 1 + 5.60T + 29T^{2} \) |
| 31 | \( 1 - 1.97T + 31T^{2} \) |
| 37 | \( 1 + 8.12T + 37T^{2} \) |
| 41 | \( 1 - 6.62T + 41T^{2} \) |
| 43 | \( 1 - 12.4T + 43T^{2} \) |
| 47 | \( 1 + 3.21T + 47T^{2} \) |
| 53 | \( 1 + 0.926T + 53T^{2} \) |
| 59 | \( 1 + 5.22T + 59T^{2} \) |
| 61 | \( 1 + 6.34T + 61T^{2} \) |
| 67 | \( 1 + 10.1T + 67T^{2} \) |
| 71 | \( 1 + 14.9T + 71T^{2} \) |
| 73 | \( 1 - 9.16T + 73T^{2} \) |
| 79 | \( 1 - 2.09T + 79T^{2} \) |
| 83 | \( 1 + 5.48T + 83T^{2} \) |
| 89 | \( 1 + 4.86T + 89T^{2} \) |
| 97 | \( 1 - 13.7T + 97T^{2} \) |
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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
−7.45072396464595042342559462606, −7.26349955636414475455080000548, −6.33170127395744362963652000499, −5.93400644321041656171802462709, −4.53860935095993663377955646090, −3.65070810778135977510316028094, −2.55151469680901172860235275210, −1.95954840682584718137398720480, −1.31665805647537722623489132928, 0,
1.31665805647537722623489132928, 1.95954840682584718137398720480, 2.55151469680901172860235275210, 3.65070810778135977510316028094, 4.53860935095993663377955646090, 5.93400644321041656171802462709, 6.33170127395744362963652000499, 7.26349955636414475455080000548, 7.45072396464595042342559462606
