| L(s) = 1 | + 0.523·3-s + 1.52·5-s − 4.20·7-s − 2.72·9-s − 4.67·11-s + 6.72·13-s + 0.798·15-s + 6.40·17-s − 2·19-s − 2.20·21-s + 2.72·23-s − 2.67·25-s − 3·27-s − 1.67·29-s + 10.9·31-s − 2.45·33-s − 6.40·35-s + 37-s + 3.52·39-s + 4.52·41-s + 3.45·43-s − 4.15·45-s + 6.20·47-s + 10.6·49-s + 3.35·51-s − 3.24·53-s − 7.12·55-s + ⋯ |
| L(s) = 1 | + 0.302·3-s + 0.681·5-s − 1.58·7-s − 0.908·9-s − 1.41·11-s + 1.86·13-s + 0.206·15-s + 1.55·17-s − 0.458·19-s − 0.480·21-s + 0.568·23-s − 0.535·25-s − 0.577·27-s − 0.311·29-s + 1.97·31-s − 0.426·33-s − 1.08·35-s + 0.164·37-s + 0.564·39-s + 0.706·41-s + 0.526·43-s − 0.619·45-s + 0.904·47-s + 1.52·49-s + 0.469·51-s − 0.446·53-s − 0.961·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.699092343\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.699092343\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 3 | \( 1 - 0.523T + 3T^{2} \) |
| 5 | \( 1 - 1.52T + 5T^{2} \) |
| 7 | \( 1 + 4.20T + 7T^{2} \) |
| 11 | \( 1 + 4.67T + 11T^{2} \) |
| 13 | \( 1 - 6.72T + 13T^{2} \) |
| 17 | \( 1 - 6.40T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 2.72T + 23T^{2} \) |
| 29 | \( 1 + 1.67T + 29T^{2} \) |
| 31 | \( 1 - 10.9T + 31T^{2} \) |
| 41 | \( 1 - 4.52T + 41T^{2} \) |
| 43 | \( 1 - 3.45T + 43T^{2} \) |
| 47 | \( 1 - 6.20T + 47T^{2} \) |
| 53 | \( 1 + 3.24T + 53T^{2} \) |
| 59 | \( 1 - 9.85T + 59T^{2} \) |
| 61 | \( 1 - 6.87T + 61T^{2} \) |
| 67 | \( 1 + 5.52T + 67T^{2} \) |
| 71 | \( 1 - 5.15T + 71T^{2} \) |
| 73 | \( 1 + 7.62T + 73T^{2} \) |
| 79 | \( 1 - 2.32T + 79T^{2} \) |
| 83 | \( 1 - 3.15T + 83T^{2} \) |
| 89 | \( 1 - 16.4T + 89T^{2} \) |
| 97 | \( 1 + 10.8T + 97T^{2} \) |
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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
−8.965822727148746006874850671214, −8.302153539675970948287014535095, −7.56385261350636035912084662316, −6.32035788274639914962643076440, −5.98875970658793967039432940362, −5.33193610866283265180108438996, −3.82932870289831882364147850074, −3.09932505753354094600048354523, −2.46468218223357102506213764549, −0.811045846678738965948624070845,
0.811045846678738965948624070845, 2.46468218223357102506213764549, 3.09932505753354094600048354523, 3.82932870289831882364147850074, 5.33193610866283265180108438996, 5.98875970658793967039432940362, 6.32035788274639914962643076440, 7.56385261350636035912084662316, 8.302153539675970948287014535095, 8.965822727148746006874850671214
