| L(s) = 1 | + 3.33·2-s + 3.11·4-s − 5·5-s + 1.26·7-s − 16.2·8-s − 16.6·10-s + 35.2·11-s − 44.9·13-s + 4.21·14-s − 79.2·16-s + 17.8·17-s + 113.·19-s − 15.5·20-s + 117.·22-s − 147.·23-s + 25·25-s − 149.·26-s + 3.93·28-s + 29·29-s + 65.4·31-s − 133.·32-s + 59.4·34-s − 6.31·35-s + 399.·37-s + 377.·38-s + 81.4·40-s + 271.·41-s + ⋯ |
| L(s) = 1 | + 1.17·2-s + 0.389·4-s − 0.447·5-s + 0.0682·7-s − 0.719·8-s − 0.527·10-s + 0.966·11-s − 0.959·13-s + 0.0804·14-s − 1.23·16-s + 0.254·17-s + 1.36·19-s − 0.174·20-s + 1.13·22-s − 1.33·23-s + 0.200·25-s − 1.13·26-s + 0.0265·28-s + 0.185·29-s + 0.379·31-s − 0.739·32-s + 0.300·34-s − 0.0305·35-s + 1.77·37-s + 1.61·38-s + 0.321·40-s + 1.03·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.192717912\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.192717912\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 29 | \( 1 - 29T \) |
| good | 2 | \( 1 - 3.33T + 8T^{2} \) |
| 7 | \( 1 - 1.26T + 343T^{2} \) |
| 11 | \( 1 - 35.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 44.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 17.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 113.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 147.T + 1.21e4T^{2} \) |
| 31 | \( 1 - 65.4T + 2.97e4T^{2} \) |
| 37 | \( 1 - 399.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 271.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 242.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 374.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 580.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 803.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 842.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 266.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 732.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 227.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 680.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 433.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.55e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.02e3T + 9.12e5T^{2} \) |
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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
−9.495771679912587781012828299385, −8.345640182999610129872150476415, −7.52557176028013931632382343705, −6.59920746594028912312128283215, −5.78543400641913069919005725892, −4.91009078558169633593412049480, −4.14960242673788528294353839877, −3.39861351121221073535096759041, −2.35550989068813499222307949455, −0.75063319581679578147575481685,
0.75063319581679578147575481685, 2.35550989068813499222307949455, 3.39861351121221073535096759041, 4.14960242673788528294353839877, 4.91009078558169633593412049480, 5.78543400641913069919005725892, 6.59920746594028912312128283215, 7.52557176028013931632382343705, 8.345640182999610129872150476415, 9.495771679912587781012828299385
