| L(s) = 1 | − 2.63·2-s − 2.71·3-s + 4.95·4-s + 2.08·5-s + 7.15·6-s + 7-s − 7.80·8-s + 4.34·9-s − 5.49·10-s + 3.60·11-s − 13.4·12-s − 2.63·14-s − 5.64·15-s + 10.6·16-s − 5.76·17-s − 11.4·18-s − 1.36·19-s + 10.3·20-s − 2.71·21-s − 9.51·22-s − 4.22·23-s + 21.1·24-s − 0.661·25-s − 3.65·27-s + 4.95·28-s − 8.29·29-s + 14.8·30-s + ⋯ |
| L(s) = 1 | − 1.86·2-s − 1.56·3-s + 2.47·4-s + 0.931·5-s + 2.91·6-s + 0.377·7-s − 2.75·8-s + 1.44·9-s − 1.73·10-s + 1.08·11-s − 3.87·12-s − 0.704·14-s − 1.45·15-s + 2.66·16-s − 1.39·17-s − 2.70·18-s − 0.312·19-s + 2.30·20-s − 0.591·21-s − 2.02·22-s − 0.881·23-s + 4.31·24-s − 0.132·25-s − 0.703·27-s + 0.936·28-s − 1.53·29-s + 2.71·30-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 | \( 1 + 2.63T + 2T^{2} \) |
| 3 | \( 1 + 2.71T + 3T^{2} \) |
| 5 | \( 1 - 2.08T + 5T^{2} \) |
| 11 | \( 1 - 3.60T + 11T^{2} \) |
| 17 | \( 1 + 5.76T + 17T^{2} \) |
| 19 | \( 1 + 1.36T + 19T^{2} \) |
| 23 | \( 1 + 4.22T + 23T^{2} \) |
| 29 | \( 1 + 8.29T + 29T^{2} \) |
| 31 | \( 1 - 0.734T + 31T^{2} \) |
| 37 | \( 1 + 2.16T + 37T^{2} \) |
| 41 | \( 1 - 2.86T + 41T^{2} \) |
| 43 | \( 1 - 4.26T + 43T^{2} \) |
| 47 | \( 1 - 8.68T + 47T^{2} \) |
| 53 | \( 1 + 10.9T + 53T^{2} \) |
| 59 | \( 1 + 6.75T + 59T^{2} \) |
| 61 | \( 1 - 10.5T + 61T^{2} \) |
| 67 | \( 1 + 4.70T + 67T^{2} \) |
| 71 | \( 1 + 13.3T + 71T^{2} \) |
| 73 | \( 1 - 4.36T + 73T^{2} \) |
| 79 | \( 1 + 1.73T + 79T^{2} \) |
| 83 | \( 1 + 5.74T + 83T^{2} \) |
| 89 | \( 1 - 8.41T + 89T^{2} \) |
| 97 | \( 1 + 6.20T + 97T^{2} \) |
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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
−9.364680015316727721996054812822, −8.883356127423995252053887956004, −7.69826908213164527815286506184, −6.81829653531997539644077215956, −6.22561177257925480897577650035, −5.64698405096049395237399495019, −4.24652080262011934969699892630, −2.17627762296277189559130789171, −1.38819368418339663339080442496, 0,
1.38819368418339663339080442496, 2.17627762296277189559130789171, 4.24652080262011934969699892630, 5.64698405096049395237399495019, 6.22561177257925480897577650035, 6.81829653531997539644077215956, 7.69826908213164527815286506184, 8.883356127423995252053887956004, 9.364680015316727721996054812822
