| L(s) = 1 | + 0.386·3-s + 0.386·5-s − 4.17·7-s − 2.85·9-s + 3.78·11-s + 13-s + 0.149·15-s − 4.49·17-s − 5.25·19-s − 1.61·21-s + 6.11·23-s − 4.85·25-s − 2.26·27-s + 8.88·29-s + 4.44·31-s + 1.46·33-s − 1.61·35-s + 3.96·37-s + 0.386·39-s − 8.11·41-s + 12.7·43-s − 1.10·45-s + 7.40·47-s + 10.4·49-s − 1.73·51-s − 9.04·53-s + 1.46·55-s + ⋯ |
| L(s) = 1 | + 0.223·3-s + 0.172·5-s − 1.57·7-s − 0.950·9-s + 1.14·11-s + 0.277·13-s + 0.0385·15-s − 1.09·17-s − 1.20·19-s − 0.352·21-s + 1.27·23-s − 0.970·25-s − 0.435·27-s + 1.64·29-s + 0.799·31-s + 0.254·33-s − 0.272·35-s + 0.651·37-s + 0.0618·39-s − 1.26·41-s + 1.94·43-s − 0.164·45-s + 1.07·47-s + 1.48·49-s − 0.243·51-s − 1.24·53-s + 0.197·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.408514623\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.408514623\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 0.386T + 3T^{2} \) |
| 5 | \( 1 - 0.386T + 5T^{2} \) |
| 7 | \( 1 + 4.17T + 7T^{2} \) |
| 11 | \( 1 - 3.78T + 11T^{2} \) |
| 17 | \( 1 + 4.49T + 17T^{2} \) |
| 19 | \( 1 + 5.25T + 19T^{2} \) |
| 23 | \( 1 - 6.11T + 23T^{2} \) |
| 29 | \( 1 - 8.88T + 29T^{2} \) |
| 31 | \( 1 - 4.44T + 31T^{2} \) |
| 37 | \( 1 - 3.96T + 37T^{2} \) |
| 41 | \( 1 + 8.11T + 41T^{2} \) |
| 43 | \( 1 - 12.7T + 43T^{2} \) |
| 47 | \( 1 - 7.40T + 47T^{2} \) |
| 53 | \( 1 + 9.04T + 53T^{2} \) |
| 59 | \( 1 + 4.56T + 59T^{2} \) |
| 61 | \( 1 + 1.33T + 61T^{2} \) |
| 67 | \( 1 - 6.67T + 67T^{2} \) |
| 71 | \( 1 - 4.59T + 71T^{2} \) |
| 73 | \( 1 - 11.6T + 73T^{2} \) |
| 79 | \( 1 - 2.57T + 79T^{2} \) |
| 83 | \( 1 + 2.67T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 - 0.237T + 97T^{2} \) |
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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
−8.895002079576958813017869447885, −8.037037517786610590415954659961, −6.79525542284478497638080360059, −6.43586115277667951784978510994, −5.93518129304408578046625182747, −4.65550632577740347138508245644, −3.84007212844696914655385706415, −3.01310984095734776275838325993, −2.26648898139298758020480520215, −0.67433731587662930792415650167,
0.67433731587662930792415650167, 2.26648898139298758020480520215, 3.01310984095734776275838325993, 3.84007212844696914655385706415, 4.65550632577740347138508245644, 5.93518129304408578046625182747, 6.43586115277667951784978510994, 6.79525542284478497638080360059, 8.037037517786610590415954659961, 8.895002079576958813017869447885
