| L(s) = 1 | − 3-s + 4·7-s + 9-s + 13-s + 3·19-s − 4·21-s − 4·23-s − 27-s − 29-s + 8·31-s + 37-s − 39-s + 41-s + 6·43-s − 11·47-s + 9·49-s + 3·53-s − 3·57-s − 10·59-s + 4·61-s + 4·63-s + 13·67-s + 4·69-s − 9·71-s + 3·79-s + 81-s + 2·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.51·7-s + 1/3·9-s + 0.277·13-s + 0.688·19-s − 0.872·21-s − 0.834·23-s − 0.192·27-s − 0.185·29-s + 1.43·31-s + 0.164·37-s − 0.160·39-s + 0.156·41-s + 0.914·43-s − 1.60·47-s + 9/7·49-s + 0.412·53-s − 0.397·57-s − 1.30·59-s + 0.512·61-s + 0.503·63-s + 1.58·67-s + 0.481·69-s − 1.06·71-s + 0.337·79-s + 1/9·81-s + 0.219·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.504288771\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.504288771\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 11 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−15.09339555993224, −14.37753459373410, −14.25309251060685, −13.46758166067936, −13.07134560092111, −12.15034913539542, −11.88053756287980, −11.43948284396602, −10.84117525059029, −10.45790093915670, −9.714674201899500, −9.236934125619086, −8.309497205182981, −8.076220395917364, −7.516421195085817, −6.778391028594407, −6.151348829356475, −5.550526821425567, −4.966092408505166, −4.478212088058883, −3.860338910721244, −2.941867870177652, −2.064379901147581, −1.415341196941911, −0.6687949435050789,
0.6687949435050789, 1.415341196941911, 2.064379901147581, 2.941867870177652, 3.860338910721244, 4.478212088058883, 4.966092408505166, 5.550526821425567, 6.151348829356475, 6.778391028594407, 7.516421195085817, 8.076220395917364, 8.309497205182981, 9.236934125619086, 9.714674201899500, 10.45790093915670, 10.84117525059029, 11.43948284396602, 11.88053756287980, 12.15034913539542, 13.07134560092111, 13.46758166067936, 14.25309251060685, 14.37753459373410, 15.09339555993224
