| L(s) = 1 | − 2.41·3-s − 0.771·5-s + 0.588·7-s + 2.82·9-s − 3.03·11-s − 6.82·13-s + 1.86·15-s + 5.66·17-s + 3.04·19-s − 1.42·21-s + 1.53·23-s − 4.40·25-s + 0.419·27-s − 5.88·29-s + 2.47·31-s + 7.32·33-s − 0.454·35-s + 4.71·37-s + 16.4·39-s − 4.06·41-s − 9.34·43-s − 2.18·45-s − 6.04·47-s − 6.65·49-s − 13.6·51-s + 0.436·53-s + 2.33·55-s + ⋯ |
| L(s) = 1 | − 1.39·3-s − 0.344·5-s + 0.222·7-s + 0.942·9-s − 0.914·11-s − 1.89·13-s + 0.480·15-s + 1.37·17-s + 0.698·19-s − 0.310·21-s + 0.319·23-s − 0.880·25-s + 0.0806·27-s − 1.09·29-s + 0.444·31-s + 1.27·33-s − 0.0767·35-s + 0.775·37-s + 2.63·39-s − 0.634·41-s − 1.42·43-s − 0.325·45-s − 0.882·47-s − 0.950·49-s − 1.91·51-s + 0.0600·53-s + 0.315·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4229292633\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4229292633\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 - T \) |
| good | 3 | \( 1 + 2.41T + 3T^{2} \) |
| 5 | \( 1 + 0.771T + 5T^{2} \) |
| 7 | \( 1 - 0.588T + 7T^{2} \) |
| 11 | \( 1 + 3.03T + 11T^{2} \) |
| 13 | \( 1 + 6.82T + 13T^{2} \) |
| 17 | \( 1 - 5.66T + 17T^{2} \) |
| 19 | \( 1 - 3.04T + 19T^{2} \) |
| 23 | \( 1 - 1.53T + 23T^{2} \) |
| 29 | \( 1 + 5.88T + 29T^{2} \) |
| 31 | \( 1 - 2.47T + 31T^{2} \) |
| 37 | \( 1 - 4.71T + 37T^{2} \) |
| 41 | \( 1 + 4.06T + 41T^{2} \) |
| 43 | \( 1 + 9.34T + 43T^{2} \) |
| 47 | \( 1 + 6.04T + 47T^{2} \) |
| 53 | \( 1 - 0.436T + 53T^{2} \) |
| 59 | \( 1 + 9.54T + 59T^{2} \) |
| 61 | \( 1 + 4.75T + 61T^{2} \) |
| 67 | \( 1 - 14.5T + 67T^{2} \) |
| 71 | \( 1 - 9.81T + 71T^{2} \) |
| 73 | \( 1 + 11.7T + 73T^{2} \) |
| 79 | \( 1 - 8.45T + 79T^{2} \) |
| 83 | \( 1 + 14.2T + 83T^{2} \) |
| 89 | \( 1 - 11.6T + 89T^{2} \) |
| 97 | \( 1 + 3.84T + 97T^{2} \) |
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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
−7.73115489353493145756620267642, −7.19324023089230377762150717617, −6.38001193327110899841036417138, −5.54083181039655485326002496800, −5.10307507593163324329583901460, −4.70999012043336935723135703196, −3.52755252097574475642571019753, −2.70184776368611531512155810637, −1.56724920602839437445517390212, −0.34681406799683837973212111042,
0.34681406799683837973212111042, 1.56724920602839437445517390212, 2.70184776368611531512155810637, 3.52755252097574475642571019753, 4.70999012043336935723135703196, 5.10307507593163324329583901460, 5.54083181039655485326002496800, 6.38001193327110899841036417138, 7.19324023089230377762150717617, 7.73115489353493145756620267642
