| L(s) = 1 | + 1.32·3-s + 2.70·5-s − 7-s − 1.24·9-s − 11-s + 13-s + 3.58·15-s + 5.60·17-s − 2.35·19-s − 1.32·21-s − 1.20·23-s + 2.30·25-s − 5.62·27-s + 6.92·29-s − 2.89·31-s − 1.32·33-s − 2.70·35-s + 8.97·37-s + 1.32·39-s + 5.99·41-s + 6.83·43-s − 3.35·45-s − 12.9·47-s + 49-s + 7.43·51-s + 11.8·53-s − 2.70·55-s + ⋯ |
| L(s) = 1 | + 0.765·3-s + 1.20·5-s − 0.377·7-s − 0.414·9-s − 0.301·11-s + 0.277·13-s + 0.924·15-s + 1.35·17-s − 0.540·19-s − 0.289·21-s − 0.250·23-s + 0.460·25-s − 1.08·27-s + 1.28·29-s − 0.519·31-s − 0.230·33-s − 0.456·35-s + 1.47·37-s + 0.212·39-s + 0.935·41-s + 1.04·43-s − 0.500·45-s − 1.88·47-s + 0.142·49-s + 1.04·51-s + 1.62·53-s − 0.364·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.303791716\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.303791716\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 1.32T + 3T^{2} \) |
| 5 | \( 1 - 2.70T + 5T^{2} \) |
| 17 | \( 1 - 5.60T + 17T^{2} \) |
| 19 | \( 1 + 2.35T + 19T^{2} \) |
| 23 | \( 1 + 1.20T + 23T^{2} \) |
| 29 | \( 1 - 6.92T + 29T^{2} \) |
| 31 | \( 1 + 2.89T + 31T^{2} \) |
| 37 | \( 1 - 8.97T + 37T^{2} \) |
| 41 | \( 1 - 5.99T + 41T^{2} \) |
| 43 | \( 1 - 6.83T + 43T^{2} \) |
| 47 | \( 1 + 12.9T + 47T^{2} \) |
| 53 | \( 1 - 11.8T + 53T^{2} \) |
| 59 | \( 1 + 2.16T + 59T^{2} \) |
| 61 | \( 1 + 3.92T + 61T^{2} \) |
| 67 | \( 1 - 12.2T + 67T^{2} \) |
| 71 | \( 1 - 9.50T + 71T^{2} \) |
| 73 | \( 1 + 4.77T + 73T^{2} \) |
| 79 | \( 1 + 3.55T + 79T^{2} \) |
| 83 | \( 1 - 4.94T + 83T^{2} \) |
| 89 | \( 1 - 3.42T + 89T^{2} \) |
| 97 | \( 1 + 5.75T + 97T^{2} \) |
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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
−8.040075689296836990318751532367, −7.20077113046266943803243762949, −6.23608096566396064797559686052, −5.88413830963414237032280740802, −5.20412002876927763058529031631, −4.17529506052460862545125536383, −3.29496316080465630036122187626, −2.64425066303251624040647697319, −2.00128758378490980896669928061, −0.868976630516406844691972516873,
0.868976630516406844691972516873, 2.00128758378490980896669928061, 2.64425066303251624040647697319, 3.29496316080465630036122187626, 4.17529506052460862545125536383, 5.20412002876927763058529031631, 5.88413830963414237032280740802, 6.23608096566396064797559686052, 7.20077113046266943803243762949, 8.040075689296836990318751532367
