| L(s) = 1 | − 1.67·3-s − 3.30·5-s + 1.48·7-s − 0.209·9-s + 2.77·11-s + 3.41·13-s + 5.51·15-s + 3.76·17-s − 0.514·19-s − 2.47·21-s + 5.89·25-s + 5.36·27-s + 7.25·29-s − 7.65·31-s − 4.63·33-s − 4.89·35-s − 3.28·37-s − 5.71·39-s − 10.7·41-s − 6.13·43-s + 0.690·45-s − 8.93·47-s − 4.79·49-s − 6.29·51-s − 11.3·53-s − 9.15·55-s + 0.859·57-s + ⋯ |
| L(s) = 1 | − 0.964·3-s − 1.47·5-s + 0.560·7-s − 0.0697·9-s + 0.836·11-s + 0.948·13-s + 1.42·15-s + 0.913·17-s − 0.118·19-s − 0.540·21-s + 1.17·25-s + 1.03·27-s + 1.34·29-s − 1.37·31-s − 0.806·33-s − 0.827·35-s − 0.539·37-s − 0.914·39-s − 1.68·41-s − 0.934·43-s + 0.102·45-s − 1.30·47-s − 0.685·49-s − 0.881·51-s − 1.56·53-s − 1.23·55-s + 0.113·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9261143505\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9261143505\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + 1.67T + 3T^{2} \) |
| 5 | \( 1 + 3.30T + 5T^{2} \) |
| 7 | \( 1 - 1.48T + 7T^{2} \) |
| 11 | \( 1 - 2.77T + 11T^{2} \) |
| 13 | \( 1 - 3.41T + 13T^{2} \) |
| 17 | \( 1 - 3.76T + 17T^{2} \) |
| 19 | \( 1 + 0.514T + 19T^{2} \) |
| 29 | \( 1 - 7.25T + 29T^{2} \) |
| 31 | \( 1 + 7.65T + 31T^{2} \) |
| 37 | \( 1 + 3.28T + 37T^{2} \) |
| 41 | \( 1 + 10.7T + 41T^{2} \) |
| 43 | \( 1 + 6.13T + 43T^{2} \) |
| 47 | \( 1 + 8.93T + 47T^{2} \) |
| 53 | \( 1 + 11.3T + 53T^{2} \) |
| 59 | \( 1 + 2.03T + 59T^{2} \) |
| 61 | \( 1 - 10.6T + 61T^{2} \) |
| 67 | \( 1 - 5.97T + 67T^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 - 10.0T + 73T^{2} \) |
| 79 | \( 1 - 7.58T + 79T^{2} \) |
| 83 | \( 1 - 8.04T + 83T^{2} \) |
| 89 | \( 1 - 15.7T + 89T^{2} \) |
| 97 | \( 1 - 7.14T + 97T^{2} \) |
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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
−8.228442073733707229918171110667, −7.85120354026309401845084914659, −6.66387243742519470071065946534, −6.45871595196300471436460549589, −5.19455154280897165572297419020, −4.86247037422835141545761296950, −3.67647383032983439744458015204, −3.39715209093422976528634776180, −1.62215532077008151110226024881, −0.60018318571631462171877453488,
0.60018318571631462171877453488, 1.62215532077008151110226024881, 3.39715209093422976528634776180, 3.67647383032983439744458015204, 4.86247037422835141545761296950, 5.19455154280897165572297419020, 6.45871595196300471436460549589, 6.66387243742519470071065946534, 7.85120354026309401845084914659, 8.228442073733707229918171110667
