| L(s) = 1 | − 1.80·2-s + 2.35·3-s + 1.26·4-s − 5-s − 4.26·6-s − 7-s + 1.33·8-s + 2.56·9-s + 1.80·10-s − 2.69·11-s + 2.97·12-s + 3.47·13-s + 1.80·14-s − 2.35·15-s − 4.93·16-s + 0.778·17-s − 4.63·18-s − 3.47·19-s − 1.26·20-s − 2.35·21-s + 4.87·22-s − 3.41·23-s + 3.14·24-s + 25-s − 6.28·26-s − 1.02·27-s − 1.26·28-s + ⋯ |
| L(s) = 1 | − 1.27·2-s + 1.36·3-s + 0.631·4-s − 0.447·5-s − 1.73·6-s − 0.377·7-s + 0.471·8-s + 0.854·9-s + 0.571·10-s − 0.813·11-s + 0.859·12-s + 0.964·13-s + 0.482·14-s − 0.609·15-s − 1.23·16-s + 0.188·17-s − 1.09·18-s − 0.797·19-s − 0.282·20-s − 0.514·21-s + 1.03·22-s − 0.711·23-s + 0.641·24-s + 0.200·25-s − 1.23·26-s − 0.197·27-s − 0.238·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 + T \) |
| good | 2 | \( 1 + 1.80T + 2T^{2} \) |
| 3 | \( 1 - 2.35T + 3T^{2} \) |
| 11 | \( 1 + 2.69T + 11T^{2} \) |
| 13 | \( 1 - 3.47T + 13T^{2} \) |
| 17 | \( 1 - 0.778T + 17T^{2} \) |
| 19 | \( 1 + 3.47T + 19T^{2} \) |
| 23 | \( 1 + 3.41T + 23T^{2} \) |
| 29 | \( 1 - 1.67T + 29T^{2} \) |
| 31 | \( 1 - 2.44T + 31T^{2} \) |
| 37 | \( 1 + 6.83T + 37T^{2} \) |
| 41 | \( 1 - 2.85T + 41T^{2} \) |
| 43 | \( 1 - 8.30T + 43T^{2} \) |
| 47 | \( 1 - 8.26T + 47T^{2} \) |
| 53 | \( 1 - 8.82T + 53T^{2} \) |
| 59 | \( 1 - 4.26T + 59T^{2} \) |
| 61 | \( 1 - 0.270T + 61T^{2} \) |
| 67 | \( 1 + 8.19T + 67T^{2} \) |
| 71 | \( 1 + 13.5T + 71T^{2} \) |
| 73 | \( 1 - 7.03T + 73T^{2} \) |
| 79 | \( 1 + 0.794T + 79T^{2} \) |
| 83 | \( 1 + 11.2T + 83T^{2} \) |
| 89 | \( 1 - 3.24T + 89T^{2} \) |
| 97 | \( 1 + 9.06T + 97T^{2} \) |
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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
−7.66086776084393272366057602218, −7.33765743053878010744771313204, −6.39265033148801030769263373352, −5.50534261362010258740286685148, −4.27809906818202968437169275880, −3.84475934058825708713025944408, −2.82470583990722507469158690580, −2.23322665848149703155607207201, −1.19009414345465362199220676175, 0,
1.19009414345465362199220676175, 2.23322665848149703155607207201, 2.82470583990722507469158690580, 3.84475934058825708713025944408, 4.27809906818202968437169275880, 5.50534261362010258740286685148, 6.39265033148801030769263373352, 7.33765743053878010744771313204, 7.66086776084393272366057602218
