| L(s) = 1 | + 1.35·3-s − 5-s − 0.648·7-s − 1.17·9-s + 3.35·11-s + 4.17·13-s − 1.35·15-s + 4.82·17-s + 6.82·19-s − 0.876·21-s + 5.52·23-s + 25-s − 5.64·27-s − 29-s − 2.82·31-s + 4.53·33-s + 0.648·35-s − 10.2·37-s + 5.64·39-s + 8.17·41-s − 5.69·43-s + 1.17·45-s − 2.64·47-s − 6.58·49-s + 6.51·51-s − 2.87·53-s − 3.35·55-s + ⋯ |
| L(s) = 1 | + 0.780·3-s − 0.447·5-s − 0.244·7-s − 0.390·9-s + 1.01·11-s + 1.15·13-s − 0.349·15-s + 1.16·17-s + 1.56·19-s − 0.191·21-s + 1.15·23-s + 0.200·25-s − 1.08·27-s − 0.185·29-s − 0.506·31-s + 0.788·33-s + 0.109·35-s − 1.68·37-s + 0.903·39-s + 1.27·41-s − 0.868·43-s + 0.174·45-s − 0.386·47-s − 0.940·49-s + 0.912·51-s − 0.395·53-s − 0.451·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 580 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 580 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.854412853\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.854412853\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 - 1.35T + 3T^{2} \) |
| 7 | \( 1 + 0.648T + 7T^{2} \) |
| 11 | \( 1 - 3.35T + 11T^{2} \) |
| 13 | \( 1 - 4.17T + 13T^{2} \) |
| 17 | \( 1 - 4.82T + 17T^{2} \) |
| 19 | \( 1 - 6.82T + 19T^{2} \) |
| 23 | \( 1 - 5.52T + 23T^{2} \) |
| 31 | \( 1 + 2.82T + 31T^{2} \) |
| 37 | \( 1 + 10.2T + 37T^{2} \) |
| 41 | \( 1 - 8.17T + 41T^{2} \) |
| 43 | \( 1 + 5.69T + 43T^{2} \) |
| 47 | \( 1 + 2.64T + 47T^{2} \) |
| 53 | \( 1 + 2.87T + 53T^{2} \) |
| 59 | \( 1 + 13.2T + 59T^{2} \) |
| 61 | \( 1 + 1.12T + 61T^{2} \) |
| 67 | \( 1 - 1.52T + 67T^{2} \) |
| 71 | \( 1 + 8.87T + 71T^{2} \) |
| 73 | \( 1 - 9.69T + 73T^{2} \) |
| 79 | \( 1 - 8.99T + 79T^{2} \) |
| 83 | \( 1 - 1.94T + 83T^{2} \) |
| 89 | \( 1 - 17.0T + 89T^{2} \) |
| 97 | \( 1 + 13.3T + 97T^{2} \) |
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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
−10.80627331304103982076436419185, −9.512161901610612002087229838363, −9.034758461647851174863220714401, −8.112373630502407402577088403141, −7.31620819985287588881130055999, −6.21548575679196691417634622475, −5.14858975311781970699947364605, −3.56028247887743070090471080749, −3.24932906971487173864314425848, −1.33674518785559824484713748693,
1.33674518785559824484713748693, 3.24932906971487173864314425848, 3.56028247887743070090471080749, 5.14858975311781970699947364605, 6.21548575679196691417634622475, 7.31620819985287588881130055999, 8.112373630502407402577088403141, 9.034758461647851174863220714401, 9.512161901610612002087229838363, 10.80627331304103982076436419185
