| L(s) = 1 | + 2.48·3-s − 2.90·5-s + 4.76·7-s + 3.18·9-s − 1.39·11-s + 5.92·13-s − 7.21·15-s − 0.806·17-s − 3.02·19-s + 11.8·21-s − 3.95·23-s + 3.41·25-s + 0.456·27-s + 1.70·29-s + 7.91·31-s − 3.47·33-s − 13.8·35-s + 0.267·37-s + 14.7·39-s + 9.75·41-s + 0.977·43-s − 9.23·45-s + 5.93·47-s + 15.7·49-s − 2.00·51-s − 10.8·53-s + 4.05·55-s + ⋯ |
| L(s) = 1 | + 1.43·3-s − 1.29·5-s + 1.80·7-s + 1.06·9-s − 0.421·11-s + 1.64·13-s − 1.86·15-s − 0.195·17-s − 0.692·19-s + 2.58·21-s − 0.825·23-s + 0.683·25-s + 0.0878·27-s + 0.316·29-s + 1.42·31-s − 0.605·33-s − 2.33·35-s + 0.0439·37-s + 2.35·39-s + 1.52·41-s + 0.149·43-s − 1.37·45-s + 0.866·47-s + 2.25·49-s − 0.280·51-s − 1.49·53-s + 0.547·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.711095614\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.711095614\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 - 2.48T + 3T^{2} \) |
| 5 | \( 1 + 2.90T + 5T^{2} \) |
| 7 | \( 1 - 4.76T + 7T^{2} \) |
| 11 | \( 1 + 1.39T + 11T^{2} \) |
| 13 | \( 1 - 5.92T + 13T^{2} \) |
| 17 | \( 1 + 0.806T + 17T^{2} \) |
| 19 | \( 1 + 3.02T + 19T^{2} \) |
| 23 | \( 1 + 3.95T + 23T^{2} \) |
| 29 | \( 1 - 1.70T + 29T^{2} \) |
| 31 | \( 1 - 7.91T + 31T^{2} \) |
| 37 | \( 1 - 0.267T + 37T^{2} \) |
| 41 | \( 1 - 9.75T + 41T^{2} \) |
| 43 | \( 1 - 0.977T + 43T^{2} \) |
| 47 | \( 1 - 5.93T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 + 0.574T + 59T^{2} \) |
| 61 | \( 1 + 4.46T + 61T^{2} \) |
| 67 | \( 1 - 15.4T + 67T^{2} \) |
| 71 | \( 1 - 8.80T + 71T^{2} \) |
| 73 | \( 1 - 8.74T + 73T^{2} \) |
| 79 | \( 1 + 14.4T + 79T^{2} \) |
| 83 | \( 1 + 3.50T + 83T^{2} \) |
| 89 | \( 1 - 16.8T + 89T^{2} \) |
| 97 | \( 1 - 2.08T + 97T^{2} \) |
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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
−7.957654492835102359750741021361, −7.69277025397606710812871061157, −6.63217425237292972335587447326, −5.71519430616192343668842644135, −4.62291513928924611993672003764, −4.17495740379549237093529342418, −3.63799866793541953762593342702, −2.66644729891640267703978682901, −1.90402722367244538780429588092, −0.931386708883764266005280005121,
0.931386708883764266005280005121, 1.90402722367244538780429588092, 2.66644729891640267703978682901, 3.63799866793541953762593342702, 4.17495740379549237093529342418, 4.62291513928924611993672003764, 5.71519430616192343668842644135, 6.63217425237292972335587447326, 7.69277025397606710812871061157, 7.957654492835102359750741021361
