| L(s) = 1 | − 0.517·2-s − 3-s − 0.732·4-s + 1.93·5-s + 0.517·6-s + 0.896·8-s + 9-s − 0.999·10-s + 0.732·12-s − 1.93·15-s + 0.267·16-s − 1.41·17-s − 0.517·18-s + 19-s − 1.41·20-s − 0.896·24-s + 2.73·25-s − 27-s + 0.999·30-s − 31-s − 1.03·32-s + 0.732·34-s − 0.732·36-s − 0.517·38-s + 1.73·40-s + 1.41·41-s + 1.93·45-s + ⋯ |
| L(s) = 1 | − 0.517·2-s − 3-s − 0.732·4-s + 1.93·5-s + 0.517·6-s + 0.896·8-s + 9-s − 0.999·10-s + 0.732·12-s − 1.93·15-s + 0.267·16-s − 1.41·17-s − 0.517·18-s + 19-s − 1.41·20-s − 0.896·24-s + 2.73·25-s − 27-s + 0.999·30-s − 31-s − 1.03·32-s + 0.732·34-s − 0.732·36-s − 0.517·38-s + 1.73·40-s + 1.41·41-s + 1.93·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1191 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1191 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7155319285\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7155319285\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 397 | \( 1 + T \) |
| good | 2 | \( 1 + 0.517T + T^{2} \) |
| 5 | \( 1 - 1.93T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + 1.41T + T^{2} \) |
| 19 | \( 1 - T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 - 1.41T + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - 0.517T + T^{2} \) |
| 59 | \( 1 - 0.517T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 1.73T + T^{2} \) |
| 71 | \( 1 + 1.41T + T^{2} \) |
| 73 | \( 1 - T + T^{2} \) |
| 79 | \( 1 - 1.73T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + 1.93T + T^{2} \) |
| 97 | \( 1 + 1.73T + T^{2} \) |
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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
−9.742521385849942497128881167825, −9.449543665643185405565127045740, −8.656474923803725509113230951063, −7.32706888227517467314807984746, −6.55060939338890357175867883898, −5.60382713566265677921246109853, −5.16644795623807581984927679287, −4.12217902226495480612529479400, −2.29162852291189213510023239844, −1.17828036851709212317224379828,
1.17828036851709212317224379828, 2.29162852291189213510023239844, 4.12217902226495480612529479400, 5.16644795623807581984927679287, 5.60382713566265677921246109853, 6.55060939338890357175867883898, 7.32706888227517467314807984746, 8.656474923803725509113230951063, 9.449543665643185405565127045740, 9.742521385849942497128881167825
