| L(s) = 1 | + 1.37·2-s − 0.118·4-s − 1.17·5-s + 0.469·7-s − 2.90·8-s − 1.60·10-s + 5.74·11-s − 1.85·13-s + 0.643·14-s − 3.74·16-s − 5.22·17-s + 2.12·19-s + 0.138·20-s + 7.88·22-s − 0.171·23-s − 3.62·25-s − 2.54·26-s − 0.0554·28-s + 6.19·29-s − 0.396·31-s + 0.667·32-s − 7.17·34-s − 0.550·35-s − 8.17·37-s + 2.91·38-s + 3.40·40-s + 12.4·41-s + ⋯ |
| L(s) = 1 | + 0.969·2-s − 0.0591·4-s − 0.524·5-s + 0.177·7-s − 1.02·8-s − 0.508·10-s + 1.73·11-s − 0.515·13-s + 0.172·14-s − 0.937·16-s − 1.26·17-s + 0.487·19-s + 0.0310·20-s + 1.68·22-s − 0.0358·23-s − 0.724·25-s − 0.500·26-s − 0.0104·28-s + 1.14·29-s − 0.0711·31-s + 0.118·32-s − 1.23·34-s − 0.0930·35-s − 1.34·37-s + 0.473·38-s + 0.538·40-s + 1.94·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4527 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4527 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 2 | \( 1 - 1.37T + 2T^{2} \) |
| 5 | \( 1 + 1.17T + 5T^{2} \) |
| 7 | \( 1 - 0.469T + 7T^{2} \) |
| 11 | \( 1 - 5.74T + 11T^{2} \) |
| 13 | \( 1 + 1.85T + 13T^{2} \) |
| 17 | \( 1 + 5.22T + 17T^{2} \) |
| 19 | \( 1 - 2.12T + 19T^{2} \) |
| 23 | \( 1 + 0.171T + 23T^{2} \) |
| 29 | \( 1 - 6.19T + 29T^{2} \) |
| 31 | \( 1 + 0.396T + 31T^{2} \) |
| 37 | \( 1 + 8.17T + 37T^{2} \) |
| 41 | \( 1 - 12.4T + 41T^{2} \) |
| 43 | \( 1 + 4.97T + 43T^{2} \) |
| 47 | \( 1 - 0.521T + 47T^{2} \) |
| 53 | \( 1 + 8.76T + 53T^{2} \) |
| 59 | \( 1 + 3.35T + 59T^{2} \) |
| 61 | \( 1 + 5.38T + 61T^{2} \) |
| 67 | \( 1 - 8.42T + 67T^{2} \) |
| 71 | \( 1 + 7.47T + 71T^{2} \) |
| 73 | \( 1 + 4.60T + 73T^{2} \) |
| 79 | \( 1 + 17.1T + 79T^{2} \) |
| 83 | \( 1 + 5.97T + 83T^{2} \) |
| 89 | \( 1 - 4.25T + 89T^{2} \) |
| 97 | \( 1 + 2.64T + 97T^{2} \) |
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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
−7.972975262418991666426886417539, −6.97622481815622805925902828005, −6.47190288482450525065008043167, −5.70906123468608132637759697913, −4.69587710615675533580244807711, −4.30957222101753598842898420690, −3.60708768073317208615678723122, −2.73605887449932749079587864037, −1.48596623188376351297092383626, 0,
1.48596623188376351297092383626, 2.73605887449932749079587864037, 3.60708768073317208615678723122, 4.30957222101753598842898420690, 4.69587710615675533580244807711, 5.70906123468608132637759697913, 6.47190288482450525065008043167, 6.97622481815622805925902828005, 7.972975262418991666426886417539
