| L(s) = 1 | − 2.22·3-s − 2.51·5-s − 2.56·7-s + 1.95·9-s + 5.69·11-s − 1.70·13-s + 5.59·15-s − 17-s + 6.20·19-s + 5.70·21-s + 3.26·23-s + 1.30·25-s + 2.32·27-s + 9.25·29-s − 2.12·31-s − 12.6·33-s + 6.43·35-s − 10.4·37-s + 3.80·39-s + 4.78·41-s − 1.10·43-s − 4.91·45-s + 2.99·47-s − 0.433·49-s + 2.22·51-s − 4.50·53-s − 14.2·55-s + ⋯ |
| L(s) = 1 | − 1.28·3-s − 1.12·5-s − 0.968·7-s + 0.652·9-s + 1.71·11-s − 0.473·13-s + 1.44·15-s − 0.242·17-s + 1.42·19-s + 1.24·21-s + 0.679·23-s + 0.261·25-s + 0.447·27-s + 1.71·29-s − 0.381·31-s − 2.20·33-s + 1.08·35-s − 1.72·37-s + 0.609·39-s + 0.747·41-s − 0.168·43-s − 0.732·45-s + 0.437·47-s − 0.0619·49-s + 0.311·51-s − 0.618·53-s − 1.92·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7021749469\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7021749469\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 + 2.22T + 3T^{2} \) |
| 5 | \( 1 + 2.51T + 5T^{2} \) |
| 7 | \( 1 + 2.56T + 7T^{2} \) |
| 11 | \( 1 - 5.69T + 11T^{2} \) |
| 13 | \( 1 + 1.70T + 13T^{2} \) |
| 19 | \( 1 - 6.20T + 19T^{2} \) |
| 23 | \( 1 - 3.26T + 23T^{2} \) |
| 29 | \( 1 - 9.25T + 29T^{2} \) |
| 31 | \( 1 + 2.12T + 31T^{2} \) |
| 37 | \( 1 + 10.4T + 37T^{2} \) |
| 41 | \( 1 - 4.78T + 41T^{2} \) |
| 43 | \( 1 + 1.10T + 43T^{2} \) |
| 47 | \( 1 - 2.99T + 47T^{2} \) |
| 53 | \( 1 + 4.50T + 53T^{2} \) |
| 61 | \( 1 - 12.3T + 61T^{2} \) |
| 67 | \( 1 + 6.10T + 67T^{2} \) |
| 71 | \( 1 + 13.6T + 71T^{2} \) |
| 73 | \( 1 + 14.8T + 73T^{2} \) |
| 79 | \( 1 + 12.8T + 79T^{2} \) |
| 83 | \( 1 + 6.77T + 83T^{2} \) |
| 89 | \( 1 - 16.5T + 89T^{2} \) |
| 97 | \( 1 - 15.8T + 97T^{2} \) |
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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
−7.46015173644626475347103483221, −7.08059440141176787747850396808, −6.45500947124253046931912110271, −5.89192753254768691523452999855, −4.98104669214789752688919501585, −4.39010740328710473147945132933, −3.55685692823463401994107584947, −2.97918218757413346337324589273, −1.33549198685540224866846424699, −0.49301015667888212065536675226,
0.49301015667888212065536675226, 1.33549198685540224866846424699, 2.97918218757413346337324589273, 3.55685692823463401994107584947, 4.39010740328710473147945132933, 4.98104669214789752688919501585, 5.89192753254768691523452999855, 6.45500947124253046931912110271, 7.08059440141176787747850396808, 7.46015173644626475347103483221
