| L(s) = 1 | − 3.24·3-s − 1.31·5-s + 7-s + 7.54·9-s + 0.394·11-s − 4.51·13-s + 4.27·15-s − 2.71·17-s + 19-s − 3.24·21-s + 3.45·23-s − 3.26·25-s − 14.7·27-s + 8.77·29-s + 4.71·31-s − 1.28·33-s − 1.31·35-s − 4.87·37-s + 14.6·39-s + 1.32·41-s + 2.75·43-s − 9.93·45-s − 8.34·47-s + 49-s + 8.80·51-s − 6.85·53-s − 0.520·55-s + ⋯ |
| L(s) = 1 | − 1.87·3-s − 0.589·5-s + 0.377·7-s + 2.51·9-s + 0.119·11-s − 1.25·13-s + 1.10·15-s − 0.657·17-s + 0.229·19-s − 0.708·21-s + 0.719·23-s − 0.652·25-s − 2.83·27-s + 1.62·29-s + 0.846·31-s − 0.223·33-s − 0.222·35-s − 0.801·37-s + 2.34·39-s + 0.206·41-s + 0.420·43-s − 1.48·45-s − 1.21·47-s + 0.142·49-s + 1.23·51-s − 0.941·53-s − 0.0701·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 3.24T + 3T^{2} \) |
| 5 | \( 1 + 1.31T + 5T^{2} \) |
| 11 | \( 1 - 0.394T + 11T^{2} \) |
| 13 | \( 1 + 4.51T + 13T^{2} \) |
| 17 | \( 1 + 2.71T + 17T^{2} \) |
| 23 | \( 1 - 3.45T + 23T^{2} \) |
| 29 | \( 1 - 8.77T + 29T^{2} \) |
| 31 | \( 1 - 4.71T + 31T^{2} \) |
| 37 | \( 1 + 4.87T + 37T^{2} \) |
| 41 | \( 1 - 1.32T + 41T^{2} \) |
| 43 | \( 1 - 2.75T + 43T^{2} \) |
| 47 | \( 1 + 8.34T + 47T^{2} \) |
| 53 | \( 1 + 6.85T + 53T^{2} \) |
| 59 | \( 1 + 2.34T + 59T^{2} \) |
| 61 | \( 1 + 4.25T + 61T^{2} \) |
| 67 | \( 1 + 6.42T + 67T^{2} \) |
| 71 | \( 1 - 1.42T + 71T^{2} \) |
| 73 | \( 1 + 12.4T + 73T^{2} \) |
| 79 | \( 1 - 5.62T + 79T^{2} \) |
| 83 | \( 1 - 16.9T + 83T^{2} \) |
| 89 | \( 1 - 6.11T + 89T^{2} \) |
| 97 | \( 1 - 0.381T + 97T^{2} \) |
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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
−7.36923693688887998729078040158, −6.57858935295094806632311276076, −6.18986458526895926371470510867, −5.18374554738584361114633853588, −4.74026255012292212337356763538, −4.34847891667157095291968282292, −3.16979934113627089261757331647, −1.95964475950331217397226553953, −0.903944353075606366179491817624, 0,
0.903944353075606366179491817624, 1.95964475950331217397226553953, 3.16979934113627089261757331647, 4.34847891667157095291968282292, 4.74026255012292212337356763538, 5.18374554738584361114633853588, 6.18986458526895926371470510867, 6.57858935295094806632311276076, 7.36923693688887998729078040158
