| L(s) = 1 | − 1.59·3-s + 2.11·5-s + 0.323·7-s − 0.457·9-s − 2.74·11-s − 4.84·13-s − 3.37·15-s + 4.66·17-s + 19-s − 0.515·21-s + 1.81·23-s − 0.515·25-s + 5.51·27-s − 3.81·29-s + 3.28·31-s + 4.37·33-s + 0.684·35-s + 7.27·37-s + 7.72·39-s + 3.37·41-s − 7.47·43-s − 0.967·45-s + 10.2·47-s − 6.89·49-s − 7.43·51-s − 1.33·53-s − 5.81·55-s + ⋯ |
| L(s) = 1 | − 0.920·3-s + 0.947·5-s + 0.122·7-s − 0.152·9-s − 0.828·11-s − 1.34·13-s − 0.871·15-s + 1.13·17-s + 0.229·19-s − 0.112·21-s + 0.377·23-s − 0.103·25-s + 1.06·27-s − 0.707·29-s + 0.590·31-s + 0.762·33-s + 0.115·35-s + 1.19·37-s + 1.23·39-s + 0.526·41-s − 1.14·43-s − 0.144·45-s + 1.49·47-s − 0.985·49-s − 1.04·51-s − 0.182·53-s − 0.784·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| good | 3 | \( 1 + 1.59T + 3T^{2} \) |
| 5 | \( 1 - 2.11T + 5T^{2} \) |
| 7 | \( 1 - 0.323T + 7T^{2} \) |
| 11 | \( 1 + 2.74T + 11T^{2} \) |
| 13 | \( 1 + 4.84T + 13T^{2} \) |
| 17 | \( 1 - 4.66T + 17T^{2} \) |
| 23 | \( 1 - 1.81T + 23T^{2} \) |
| 29 | \( 1 + 3.81T + 29T^{2} \) |
| 31 | \( 1 - 3.28T + 31T^{2} \) |
| 37 | \( 1 - 7.27T + 37T^{2} \) |
| 41 | \( 1 - 3.37T + 41T^{2} \) |
| 43 | \( 1 + 7.47T + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 + 1.33T + 53T^{2} \) |
| 59 | \( 1 - 5.58T + 59T^{2} \) |
| 61 | \( 1 - 2.74T + 61T^{2} \) |
| 67 | \( 1 - 5.70T + 67T^{2} \) |
| 71 | \( 1 + 0.576T + 71T^{2} \) |
| 73 | \( 1 + 12.9T + 73T^{2} \) |
| 83 | \( 1 + 9.16T + 83T^{2} \) |
| 89 | \( 1 + 5.29T + 89T^{2} \) |
| 97 | \( 1 + 13.8T + 97T^{2} \) |
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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
−7.65833063639738885074090377099, −6.95684550962151102258263287847, −6.10975068146688419891600751529, −5.42271534878141131969316529329, −5.25068103208726375639070981906, −4.29133501313412926771524318825, −2.96671573328564682029309008644, −2.39770786283067941414589176733, −1.21016927207050466288531287407, 0,
1.21016927207050466288531287407, 2.39770786283067941414589176733, 2.96671573328564682029309008644, 4.29133501313412926771524318825, 5.25068103208726375639070981906, 5.42271534878141131969316529329, 6.10975068146688419891600751529, 6.95684550962151102258263287847, 7.65833063639738885074090377099
