| L(s) = 1 | + 1.09·2-s − 0.809·4-s − 0.395·5-s + 3.35·7-s − 3.06·8-s − 0.431·10-s − 5.32·11-s − 5.62·13-s + 3.66·14-s − 1.72·16-s − 2.80·17-s + 3.32·19-s + 0.320·20-s − 5.80·22-s + 3.48·23-s − 4.84·25-s − 6.13·26-s − 2.71·28-s + 1.90·29-s − 9.80·31-s + 4.24·32-s − 3.05·34-s − 1.32·35-s + 5.47·37-s + 3.62·38-s + 1.21·40-s − 2.35·41-s + ⋯ |
| L(s) = 1 | + 0.771·2-s − 0.404·4-s − 0.177·5-s + 1.26·7-s − 1.08·8-s − 0.136·10-s − 1.60·11-s − 1.55·13-s + 0.978·14-s − 0.431·16-s − 0.680·17-s + 0.763·19-s + 0.0716·20-s − 1.23·22-s + 0.726·23-s − 0.968·25-s − 1.20·26-s − 0.513·28-s + 0.353·29-s − 1.76·31-s + 0.750·32-s − 0.524·34-s − 0.224·35-s + 0.899·37-s + 0.588·38-s + 0.191·40-s − 0.367·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 127 | \( 1 - T \) |
| good | 2 | \( 1 - 1.09T + 2T^{2} \) |
| 5 | \( 1 + 0.395T + 5T^{2} \) |
| 7 | \( 1 - 3.35T + 7T^{2} \) |
| 11 | \( 1 + 5.32T + 11T^{2} \) |
| 13 | \( 1 + 5.62T + 13T^{2} \) |
| 17 | \( 1 + 2.80T + 17T^{2} \) |
| 19 | \( 1 - 3.32T + 19T^{2} \) |
| 23 | \( 1 - 3.48T + 23T^{2} \) |
| 29 | \( 1 - 1.90T + 29T^{2} \) |
| 31 | \( 1 + 9.80T + 31T^{2} \) |
| 37 | \( 1 - 5.47T + 37T^{2} \) |
| 41 | \( 1 + 2.35T + 41T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 + 11.5T + 47T^{2} \) |
| 53 | \( 1 - 3.28T + 53T^{2} \) |
| 59 | \( 1 - 5.59T + 59T^{2} \) |
| 61 | \( 1 - 0.311T + 61T^{2} \) |
| 67 | \( 1 - 9.19T + 67T^{2} \) |
| 71 | \( 1 + 2.97T + 71T^{2} \) |
| 73 | \( 1 - 8.53T + 73T^{2} \) |
| 79 | \( 1 + 9.14T + 79T^{2} \) |
| 83 | \( 1 + 11.0T + 83T^{2} \) |
| 89 | \( 1 + 0.870T + 89T^{2} \) |
| 97 | \( 1 + 8.65T + 97T^{2} \) |
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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
−9.480959964874566829683396163998, −8.381923378994013926645517023972, −7.81867744962323334674210338630, −6.96496178454739834181493967154, −5.41072276317893606328802170840, −5.14231935271408438771989152148, −4.41596418462331074018381342322, −3.14418717730119476932277838646, −2.11326410019318090872751349439, 0,
2.11326410019318090872751349439, 3.14418717730119476932277838646, 4.41596418462331074018381342322, 5.14231935271408438771989152148, 5.41072276317893606328802170840, 6.96496178454739834181493967154, 7.81867744962323334674210338630, 8.381923378994013926645517023972, 9.480959964874566829683396163998
