| L(s) = 1 | − 1.10·2-s + 1.33·3-s − 0.776·4-s + 1.90·5-s − 1.47·6-s + 7-s + 3.07·8-s − 1.22·9-s − 2.10·10-s − 6.49·11-s − 1.03·12-s − 1.10·14-s + 2.54·15-s − 1.84·16-s − 7.14·17-s + 1.35·18-s + 4.93·19-s − 1.48·20-s + 1.33·21-s + 7.18·22-s − 6.24·23-s + 4.09·24-s − 1.36·25-s − 5.62·27-s − 0.776·28-s − 0.505·29-s − 2.81·30-s + ⋯ |
| L(s) = 1 | − 0.781·2-s + 0.768·3-s − 0.388·4-s + 0.853·5-s − 0.601·6-s + 0.377·7-s + 1.08·8-s − 0.408·9-s − 0.667·10-s − 1.95·11-s − 0.298·12-s − 0.295·14-s + 0.656·15-s − 0.460·16-s − 1.73·17-s + 0.319·18-s + 1.13·19-s − 0.331·20-s + 0.290·21-s + 1.53·22-s − 1.30·23-s + 0.834·24-s − 0.272·25-s − 1.08·27-s − 0.146·28-s − 0.0939·29-s − 0.513·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.10T + 2T^{2} \) |
| 3 | \( 1 - 1.33T + 3T^{2} \) |
| 5 | \( 1 - 1.90T + 5T^{2} \) |
| 11 | \( 1 + 6.49T + 11T^{2} \) |
| 17 | \( 1 + 7.14T + 17T^{2} \) |
| 19 | \( 1 - 4.93T + 19T^{2} \) |
| 23 | \( 1 + 6.24T + 23T^{2} \) |
| 29 | \( 1 + 0.505T + 29T^{2} \) |
| 31 | \( 1 + 5.66T + 31T^{2} \) |
| 37 | \( 1 + 0.0345T + 37T^{2} \) |
| 41 | \( 1 - 2.58T + 41T^{2} \) |
| 43 | \( 1 - 0.374T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 - 2.66T + 53T^{2} \) |
| 59 | \( 1 + 1.28T + 59T^{2} \) |
| 61 | \( 1 + 5.66T + 61T^{2} \) |
| 67 | \( 1 - 12.8T + 67T^{2} \) |
| 71 | \( 1 + 9.38T + 71T^{2} \) |
| 73 | \( 1 + 4.37T + 73T^{2} \) |
| 79 | \( 1 - 0.870T + 79T^{2} \) |
| 83 | \( 1 + 7.76T + 83T^{2} \) |
| 89 | \( 1 + 9.16T + 89T^{2} \) |
| 97 | \( 1 + 1.40T + 97T^{2} \) |
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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
−9.248538203369472238227704474964, −8.646920459311569198565151458251, −7.892549271444566297118223283598, −7.36448018407342185660003055533, −5.81754150668901226169490499623, −5.19394652366583432137449245271, −4.07490714585619872008789571773, −2.62496170229204047211364004208, −1.93886454377728794391497652841, 0,
1.93886454377728794391497652841, 2.62496170229204047211364004208, 4.07490714585619872008789571773, 5.19394652366583432137449245271, 5.81754150668901226169490499623, 7.36448018407342185660003055533, 7.892549271444566297118223283598, 8.646920459311569198565151458251, 9.248538203369472238227704474964
