| L(s) = 1 | − 1.58·3-s + 3.28·5-s − 0.477·9-s − 11-s − 4.98·13-s − 5.22·15-s − 3.69·17-s + 5.69·19-s + 0.699·23-s + 5.81·25-s + 5.52·27-s + 7.68·29-s + 10.5·31-s + 1.58·33-s − 10.5·37-s + 7.92·39-s + 7.81·41-s + 1.63·43-s − 1.56·45-s + 2.30·47-s + 5.87·51-s − 3.21·53-s − 3.28·55-s − 9.05·57-s + 7.76·59-s + 4.95·61-s − 16.3·65-s + ⋯ |
| L(s) = 1 | − 0.917·3-s + 1.47·5-s − 0.159·9-s − 0.301·11-s − 1.38·13-s − 1.34·15-s − 0.897·17-s + 1.30·19-s + 0.145·23-s + 1.16·25-s + 1.06·27-s + 1.42·29-s + 1.88·31-s + 0.276·33-s − 1.73·37-s + 1.26·39-s + 1.21·41-s + 0.249·43-s − 0.233·45-s + 0.335·47-s + 0.822·51-s − 0.440·53-s − 0.443·55-s − 1.19·57-s + 1.01·59-s + 0.634·61-s − 2.03·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.466913727\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.466913727\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + 1.58T + 3T^{2} \) |
| 5 | \( 1 - 3.28T + 5T^{2} \) |
| 13 | \( 1 + 4.98T + 13T^{2} \) |
| 17 | \( 1 + 3.69T + 17T^{2} \) |
| 19 | \( 1 - 5.69T + 19T^{2} \) |
| 23 | \( 1 - 0.699T + 23T^{2} \) |
| 29 | \( 1 - 7.68T + 29T^{2} \) |
| 31 | \( 1 - 10.5T + 31T^{2} \) |
| 37 | \( 1 + 10.5T + 37T^{2} \) |
| 41 | \( 1 - 7.81T + 41T^{2} \) |
| 43 | \( 1 - 1.63T + 43T^{2} \) |
| 47 | \( 1 - 2.30T + 47T^{2} \) |
| 53 | \( 1 + 3.21T + 53T^{2} \) |
| 59 | \( 1 - 7.76T + 59T^{2} \) |
| 61 | \( 1 - 4.95T + 61T^{2} \) |
| 67 | \( 1 + 1.62T + 67T^{2} \) |
| 71 | \( 1 + 2.30T + 71T^{2} \) |
| 73 | \( 1 - 5.41T + 73T^{2} \) |
| 79 | \( 1 + 2.73T + 79T^{2} \) |
| 83 | \( 1 - 6.65T + 83T^{2} \) |
| 89 | \( 1 + 12.8T + 89T^{2} \) |
| 97 | \( 1 - 18.6T + 97T^{2} \) |
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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
−9.224502527968877552059487738689, −8.427793985647705734706616561920, −7.27698145175473610896998241982, −6.56019951359375259206050961243, −5.89471029131576062621962691521, −5.11959625964079445343738186527, −4.70155178357072065026032667729, −2.93033685130377746175972659909, −2.24538357595899435226332145480, −0.826657008940765515896959263898,
0.826657008940765515896959263898, 2.24538357595899435226332145480, 2.93033685130377746175972659909, 4.70155178357072065026032667729, 5.11959625964079445343738186527, 5.89471029131576062621962691521, 6.56019951359375259206050961243, 7.27698145175473610896998241982, 8.427793985647705734706616561920, 9.224502527968877552059487738689
