| L(s) = 1 | − 2.74·2-s + 1.36·3-s + 5.51·4-s + 0.741·5-s − 3.74·6-s − 7-s − 9.63·8-s − 1.13·9-s − 2.03·10-s − 1.36·11-s + 7.52·12-s + 2.74·14-s + 1.01·15-s + 15.3·16-s − 4.14·17-s + 3.11·18-s + 7.26·19-s + 4.08·20-s − 1.36·21-s + 3.74·22-s − 2.33·23-s − 13.1·24-s − 4.45·25-s − 5.64·27-s − 5.51·28-s − 0.407·29-s − 2.77·30-s + ⋯ |
| L(s) = 1 | − 1.93·2-s + 0.787·3-s + 2.75·4-s + 0.331·5-s − 1.52·6-s − 0.377·7-s − 3.40·8-s − 0.379·9-s − 0.642·10-s − 0.411·11-s + 2.17·12-s + 0.732·14-s + 0.261·15-s + 3.84·16-s − 1.00·17-s + 0.734·18-s + 1.66·19-s + 0.913·20-s − 0.297·21-s + 0.797·22-s − 0.486·23-s − 2.68·24-s − 0.890·25-s − 1.08·27-s − 1.04·28-s − 0.0756·29-s − 0.506·30-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 | \( 1 + 2.74T + 2T^{2} \) |
| 3 | \( 1 - 1.36T + 3T^{2} \) |
| 5 | \( 1 - 0.741T + 5T^{2} \) |
| 11 | \( 1 + 1.36T + 11T^{2} \) |
| 17 | \( 1 + 4.14T + 17T^{2} \) |
| 19 | \( 1 - 7.26T + 19T^{2} \) |
| 23 | \( 1 + 2.33T + 23T^{2} \) |
| 29 | \( 1 + 0.407T + 29T^{2} \) |
| 31 | \( 1 + 2.77T + 31T^{2} \) |
| 37 | \( 1 + 6.10T + 37T^{2} \) |
| 41 | \( 1 - 1.25T + 41T^{2} \) |
| 43 | \( 1 + 1.74T + 43T^{2} \) |
| 47 | \( 1 + 5.85T + 47T^{2} \) |
| 53 | \( 1 - 4.56T + 53T^{2} \) |
| 59 | \( 1 + 10.9T + 59T^{2} \) |
| 61 | \( 1 - 6.52T + 61T^{2} \) |
| 67 | \( 1 + 13.7T + 67T^{2} \) |
| 71 | \( 1 + 4.81T + 71T^{2} \) |
| 73 | \( 1 - 6.06T + 73T^{2} \) |
| 79 | \( 1 + 9.12T + 79T^{2} \) |
| 83 | \( 1 - 11.7T + 83T^{2} \) |
| 89 | \( 1 + 1.76T + 89T^{2} \) |
| 97 | \( 1 - 9.53T + 97T^{2} \) |
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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
−9.273591977132472940219163615926, −8.728408989382655797451736098525, −7.88025367250662171052104722002, −7.34217309350719181582267076075, −6.35660735153892468669103418353, −5.49537337371730705192760584866, −3.45830498819834771403943812525, −2.61042207774998524409899637528, −1.69855340035475076471400518124, 0,
1.69855340035475076471400518124, 2.61042207774998524409899637528, 3.45830498819834771403943812525, 5.49537337371730705192760584866, 6.35660735153892468669103418353, 7.34217309350719181582267076075, 7.88025367250662171052104722002, 8.728408989382655797451736098525, 9.273591977132472940219163615926
