| L(s) = 1 | + 2.66·3-s − 3.79·5-s + 4.12·9-s − 4.34·11-s + 7.01·13-s − 10.1·15-s − 1.38·17-s − 1.56·19-s − 1.74·23-s + 9.39·25-s + 3.00·27-s + 7.48·29-s + 3.05·31-s − 11.5·33-s + 4.73·37-s + 18.7·39-s + 9.91·41-s + 10.8·43-s − 15.6·45-s + 0.965·47-s − 3.68·51-s + 12.9·53-s + 16.4·55-s − 4.16·57-s − 1.06·59-s − 3.34·61-s − 26.6·65-s + ⋯ |
| L(s) = 1 | + 1.54·3-s − 1.69·5-s + 1.37·9-s − 1.30·11-s + 1.94·13-s − 2.61·15-s − 0.335·17-s − 0.358·19-s − 0.364·23-s + 1.87·25-s + 0.578·27-s + 1.38·29-s + 0.548·31-s − 2.01·33-s + 0.779·37-s + 2.99·39-s + 1.54·41-s + 1.65·43-s − 2.33·45-s + 0.140·47-s − 0.516·51-s + 1.77·53-s + 2.22·55-s − 0.551·57-s − 0.138·59-s − 0.428·61-s − 3.30·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.308782914\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.308782914\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 2.66T + 3T^{2} \) |
| 5 | \( 1 + 3.79T + 5T^{2} \) |
| 11 | \( 1 + 4.34T + 11T^{2} \) |
| 13 | \( 1 - 7.01T + 13T^{2} \) |
| 17 | \( 1 + 1.38T + 17T^{2} \) |
| 19 | \( 1 + 1.56T + 19T^{2} \) |
| 23 | \( 1 + 1.74T + 23T^{2} \) |
| 29 | \( 1 - 7.48T + 29T^{2} \) |
| 31 | \( 1 - 3.05T + 31T^{2} \) |
| 37 | \( 1 - 4.73T + 37T^{2} \) |
| 41 | \( 1 - 9.91T + 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 - 0.965T + 47T^{2} \) |
| 53 | \( 1 - 12.9T + 53T^{2} \) |
| 59 | \( 1 + 1.06T + 59T^{2} \) |
| 61 | \( 1 + 3.34T + 61T^{2} \) |
| 67 | \( 1 + 7.24T + 67T^{2} \) |
| 71 | \( 1 - 4.05T + 71T^{2} \) |
| 73 | \( 1 - 3.55T + 73T^{2} \) |
| 79 | \( 1 - 8.44T + 79T^{2} \) |
| 83 | \( 1 + 10.1T + 83T^{2} \) |
| 89 | \( 1 + 7.69T + 89T^{2} \) |
| 97 | \( 1 + 6.13T + 97T^{2} \) |
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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
−8.613099555413578648212712565296, −8.006384838470289745013654525970, −7.80597086547516345431779510660, −6.83349621237594482981706922800, −5.78468138422978543200646085210, −4.32153963200447032957069225182, −4.06684400297197771045393951471, −3.10225805628734455411933291859, −2.49749589647128930053506282799, −0.887666526384914084305906130955,
0.887666526384914084305906130955, 2.49749589647128930053506282799, 3.10225805628734455411933291859, 4.06684400297197771045393951471, 4.32153963200447032957069225182, 5.78468138422978543200646085210, 6.83349621237594482981706922800, 7.80597086547516345431779510660, 8.006384838470289745013654525970, 8.613099555413578648212712565296
