| L(s) = 1 | − 3-s + 2.90·5-s − 2.88·7-s + 9-s − 3.21·11-s − 2.90·15-s + 5.12·17-s − 1.45·19-s + 2.88·21-s − 4.09·23-s + 3.42·25-s − 27-s + 3.45·29-s + 7.55·31-s + 3.21·33-s − 8.36·35-s − 11.4·37-s + 7.52·41-s + 6.51·43-s + 2.90·45-s − 9.09·47-s + 1.30·49-s − 5.12·51-s + 5.65·53-s − 9.31·55-s + 1.45·57-s + 7.21·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.29·5-s − 1.08·7-s + 0.333·9-s − 0.968·11-s − 0.749·15-s + 1.24·17-s − 0.333·19-s + 0.628·21-s − 0.853·23-s + 0.684·25-s − 0.192·27-s + 0.641·29-s + 1.35·31-s + 0.559·33-s − 1.41·35-s − 1.87·37-s + 1.17·41-s + 0.994·43-s + 0.432·45-s − 1.32·47-s + 0.186·49-s − 0.718·51-s + 0.776·53-s − 1.25·55-s + 0.192·57-s + 0.939·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.545144864\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.545144864\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 2.90T + 5T^{2} \) |
| 7 | \( 1 + 2.88T + 7T^{2} \) |
| 11 | \( 1 + 3.21T + 11T^{2} \) |
| 17 | \( 1 - 5.12T + 17T^{2} \) |
| 19 | \( 1 + 1.45T + 19T^{2} \) |
| 23 | \( 1 + 4.09T + 23T^{2} \) |
| 29 | \( 1 - 3.45T + 29T^{2} \) |
| 31 | \( 1 - 7.55T + 31T^{2} \) |
| 37 | \( 1 + 11.4T + 37T^{2} \) |
| 41 | \( 1 - 7.52T + 41T^{2} \) |
| 43 | \( 1 - 6.51T + 43T^{2} \) |
| 47 | \( 1 + 9.09T + 47T^{2} \) |
| 53 | \( 1 - 5.65T + 53T^{2} \) |
| 59 | \( 1 - 7.21T + 59T^{2} \) |
| 61 | \( 1 + 1.55T + 61T^{2} \) |
| 67 | \( 1 + 5.43T + 67T^{2} \) |
| 71 | \( 1 - 15.2T + 71T^{2} \) |
| 73 | \( 1 - 15.4T + 73T^{2} \) |
| 79 | \( 1 - 5.94T + 79T^{2} \) |
| 83 | \( 1 - 8.20T + 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 + 19.0T + 97T^{2} \) |
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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
−8.414760237916384879044250035207, −7.68260697510574790206254701760, −6.64782940278934322008297658412, −6.24246821812617071047427311182, −5.52767194461728683453869422806, −4.99672958183760717959219269360, −3.77471390031117771324520157910, −2.84072333602776856112348275374, −2.01881155979636331000091064748, −0.71827938976771118462923943964,
0.71827938976771118462923943964, 2.01881155979636331000091064748, 2.84072333602776856112348275374, 3.77471390031117771324520157910, 4.99672958183760717959219269360, 5.52767194461728683453869422806, 6.24246821812617071047427311182, 6.64782940278934322008297658412, 7.68260697510574790206254701760, 8.414760237916384879044250035207
