| L(s) = 1 | + 1.61·3-s + 10.5·5-s − 13.7·7-s − 24.3·9-s − 30.2·11-s − 19.9·13-s + 16.9·15-s + 105.·17-s − 79.5·19-s − 22.1·21-s + 14.8·23-s − 14.0·25-s − 82.9·27-s − 29·29-s + 61.9·31-s − 48.8·33-s − 144.·35-s − 6.43·37-s − 32.1·39-s + 351.·41-s + 473.·43-s − 256.·45-s − 492.·47-s − 154.·49-s + 170.·51-s + 668.·53-s − 319.·55-s + ⋯ |
| L(s) = 1 | + 0.310·3-s + 0.942·5-s − 0.741·7-s − 0.903·9-s − 0.830·11-s − 0.425·13-s + 0.292·15-s + 1.50·17-s − 0.960·19-s − 0.230·21-s + 0.134·23-s − 0.112·25-s − 0.591·27-s − 0.185·29-s + 0.358·31-s − 0.257·33-s − 0.698·35-s − 0.0285·37-s − 0.132·39-s + 1.33·41-s + 1.67·43-s − 0.851·45-s − 1.52·47-s − 0.449·49-s + 0.467·51-s + 1.73·53-s − 0.782·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.990541191\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.990541191\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 + 29T \) |
| good | 3 | \( 1 - 1.61T + 27T^{2} \) |
| 5 | \( 1 - 10.5T + 125T^{2} \) |
| 7 | \( 1 + 13.7T + 343T^{2} \) |
| 11 | \( 1 + 30.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 19.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 105.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 79.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 14.8T + 1.21e4T^{2} \) |
| 31 | \( 1 - 61.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 6.43T + 5.06e4T^{2} \) |
| 41 | \( 1 - 351.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 473.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 492.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 668.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 12.7T + 2.05e5T^{2} \) |
| 61 | \( 1 + 80.8T + 2.26e5T^{2} \) |
| 67 | \( 1 - 529.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 211.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 627.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 915.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 411.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.37e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 649.T + 9.12e5T^{2} \) |
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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
−8.980701345513606127339276753210, −8.102216495422621564639093680744, −7.42609152638054784383462134122, −6.25478800824597654193929642235, −5.78954923032951066165424080722, −5.00902384104932179418242260469, −3.68567846635537861747571170051, −2.77873405345727956247149872975, −2.14023402187025880037822955948, −0.61951301601853404139701614756,
0.61951301601853404139701614756, 2.14023402187025880037822955948, 2.77873405345727956247149872975, 3.68567846635537861747571170051, 5.00902384104932179418242260469, 5.78954923032951066165424080722, 6.25478800824597654193929642235, 7.42609152638054784383462134122, 8.102216495422621564639093680744, 8.980701345513606127339276753210
