| L(s) = 1 | − 1.06·3-s + 0.715·5-s − 1.86·9-s − 0.203·11-s + 4.07·13-s − 0.763·15-s − 1.49·17-s + 1.74·19-s − 4.09·23-s − 4.48·25-s + 5.18·27-s − 5.13·29-s + 3.89·31-s + 0.216·33-s + 9.19·37-s − 4.34·39-s + 41-s + 1.74·43-s − 1.33·45-s − 6.43·47-s + 1.59·51-s + 6.61·53-s − 0.145·55-s − 1.86·57-s − 12.8·59-s − 14.5·61-s + 2.91·65-s + ⋯ |
| L(s) = 1 | − 0.615·3-s + 0.319·5-s − 0.620·9-s − 0.0612·11-s + 1.13·13-s − 0.197·15-s − 0.363·17-s + 0.400·19-s − 0.854·23-s − 0.897·25-s + 0.998·27-s − 0.953·29-s + 0.699·31-s + 0.0377·33-s + 1.51·37-s − 0.696·39-s + 0.156·41-s + 0.266·43-s − 0.198·45-s − 0.938·47-s + 0.223·51-s + 0.908·53-s − 0.0196·55-s − 0.246·57-s − 1.67·59-s − 1.85·61-s + 0.361·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 3 | \( 1 + 1.06T + 3T^{2} \) |
| 5 | \( 1 - 0.715T + 5T^{2} \) |
| 11 | \( 1 + 0.203T + 11T^{2} \) |
| 13 | \( 1 - 4.07T + 13T^{2} \) |
| 17 | \( 1 + 1.49T + 17T^{2} \) |
| 19 | \( 1 - 1.74T + 19T^{2} \) |
| 23 | \( 1 + 4.09T + 23T^{2} \) |
| 29 | \( 1 + 5.13T + 29T^{2} \) |
| 31 | \( 1 - 3.89T + 31T^{2} \) |
| 37 | \( 1 - 9.19T + 37T^{2} \) |
| 43 | \( 1 - 1.74T + 43T^{2} \) |
| 47 | \( 1 + 6.43T + 47T^{2} \) |
| 53 | \( 1 - 6.61T + 53T^{2} \) |
| 59 | \( 1 + 12.8T + 59T^{2} \) |
| 61 | \( 1 + 14.5T + 61T^{2} \) |
| 67 | \( 1 + 0.105T + 67T^{2} \) |
| 71 | \( 1 - 14.3T + 71T^{2} \) |
| 73 | \( 1 + 13.2T + 73T^{2} \) |
| 79 | \( 1 - 6.88T + 79T^{2} \) |
| 83 | \( 1 - 2.43T + 83T^{2} \) |
| 89 | \( 1 - 4.37T + 89T^{2} \) |
| 97 | \( 1 + 11.1T + 97T^{2} \) |
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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
−7.62329153881514123827132314481, −6.45594661996793375362727006085, −6.10513095286053767694389252338, −5.60541984302721372226132781187, −4.72969184997188514739179447063, −3.95213346096842391159777375342, −3.11267644385191457157338042235, −2.18315797271928268118371144179, −1.18217245268190877834775536750, 0,
1.18217245268190877834775536750, 2.18315797271928268118371144179, 3.11267644385191457157338042235, 3.95213346096842391159777375342, 4.72969184997188514739179447063, 5.60541984302721372226132781187, 6.10513095286053767694389252338, 6.45594661996793375362727006085, 7.62329153881514123827132314481
