| L(s) = 1 | − 2-s − 3-s + 4-s + 4.30·5-s + 6-s − 3.32·7-s − 8-s + 9-s − 4.30·10-s − 2.99·11-s − 12-s − 13-s + 3.32·14-s − 4.30·15-s + 16-s + 7.37·17-s − 18-s + 1.56·19-s + 4.30·20-s + 3.32·21-s + 2.99·22-s − 3.98·23-s + 24-s + 13.5·25-s + 26-s − 27-s − 3.32·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.92·5-s + 0.408·6-s − 1.25·7-s − 0.353·8-s + 0.333·9-s − 1.36·10-s − 0.901·11-s − 0.288·12-s − 0.277·13-s + 0.888·14-s − 1.11·15-s + 0.250·16-s + 1.78·17-s − 0.235·18-s + 0.358·19-s + 0.962·20-s + 0.725·21-s + 0.637·22-s − 0.831·23-s + 0.204·24-s + 2.70·25-s + 0.196·26-s − 0.192·27-s − 0.628·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
| good | 5 | \( 1 - 4.30T + 5T^{2} \) |
| 7 | \( 1 + 3.32T + 7T^{2} \) |
| 11 | \( 1 + 2.99T + 11T^{2} \) |
| 17 | \( 1 - 7.37T + 17T^{2} \) |
| 19 | \( 1 - 1.56T + 19T^{2} \) |
| 23 | \( 1 + 3.98T + 23T^{2} \) |
| 29 | \( 1 + 5.10T + 29T^{2} \) |
| 31 | \( 1 + 6.12T + 31T^{2} \) |
| 37 | \( 1 + 5.21T + 37T^{2} \) |
| 41 | \( 1 + 1.29T + 41T^{2} \) |
| 43 | \( 1 + 2.11T + 43T^{2} \) |
| 47 | \( 1 - 10.0T + 47T^{2} \) |
| 53 | \( 1 + 8.84T + 53T^{2} \) |
| 59 | \( 1 - 3.83T + 59T^{2} \) |
| 61 | \( 1 - 4.47T + 61T^{2} \) |
| 67 | \( 1 - 1.76T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 - 9.29T + 73T^{2} \) |
| 79 | \( 1 - 12.4T + 79T^{2} \) |
| 83 | \( 1 + 7.29T + 83T^{2} \) |
| 89 | \( 1 - 0.220T + 89T^{2} \) |
| 97 | \( 1 + 10.6T + 97T^{2} \) |
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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
−7.35139654022612613480746107146, −6.77032414826438908159711159755, −6.02053216455515234947441449670, −5.53958804981065837499139902609, −5.22507683509697343436872180748, −3.64861206174391463803602758682, −2.87601607983744077863447590557, −2.08376673136693648188962794780, −1.23103504121739320786485153486, 0,
1.23103504121739320786485153486, 2.08376673136693648188962794780, 2.87601607983744077863447590557, 3.64861206174391463803602758682, 5.22507683509697343436872180748, 5.53958804981065837499139902609, 6.02053216455515234947441449670, 6.77032414826438908159711159755, 7.35139654022612613480746107146
