| L(s) = 1 | + 3-s + 1.23·5-s + 7-s − 2·9-s + 4.47·11-s − 4.23·13-s + 1.23·15-s + 2.76·19-s + 21-s + 23-s − 3.47·25-s − 5·27-s + 7.47·29-s + 9·31-s + 4.47·33-s + 1.23·35-s + 7.70·37-s − 4.23·39-s + 2.23·41-s + 6.47·43-s − 2.47·45-s − 5.47·47-s + 49-s + 6.76·53-s + 5.52·55-s + 2.76·57-s + 10.4·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.552·5-s + 0.377·7-s − 0.666·9-s + 1.34·11-s − 1.17·13-s + 0.319·15-s + 0.634·19-s + 0.218·21-s + 0.208·23-s − 0.694·25-s − 0.962·27-s + 1.38·29-s + 1.61·31-s + 0.778·33-s + 0.208·35-s + 1.26·37-s − 0.678·39-s + 0.349·41-s + 0.986·43-s − 0.368·45-s − 0.798·47-s + 0.142·49-s + 0.929·53-s + 0.745·55-s + 0.366·57-s + 1.36·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.621777556\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.621777556\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - T + 3T^{2} \) |
| 5 | \( 1 - 1.23T + 5T^{2} \) |
| 11 | \( 1 - 4.47T + 11T^{2} \) |
| 13 | \( 1 + 4.23T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 2.76T + 19T^{2} \) |
| 29 | \( 1 - 7.47T + 29T^{2} \) |
| 31 | \( 1 - 9T + 31T^{2} \) |
| 37 | \( 1 - 7.70T + 37T^{2} \) |
| 41 | \( 1 - 2.23T + 41T^{2} \) |
| 43 | \( 1 - 6.47T + 43T^{2} \) |
| 47 | \( 1 + 5.47T + 47T^{2} \) |
| 53 | \( 1 - 6.76T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 + 13.4T + 61T^{2} \) |
| 67 | \( 1 + 10.1T + 67T^{2} \) |
| 71 | \( 1 - 5.76T + 71T^{2} \) |
| 73 | \( 1 - 6.70T + 73T^{2} \) |
| 79 | \( 1 - 2.76T + 79T^{2} \) |
| 83 | \( 1 - 2.47T + 83T^{2} \) |
| 89 | \( 1 + 8.94T + 89T^{2} \) |
| 97 | \( 1 + 9.70T + 97T^{2} \) |
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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
−8.944380108876022801650734435709, −8.173820523511472007915594724913, −7.48985068612798017467480657882, −6.52903438750615800839683333880, −5.87922716924499820150219554383, −4.89403119134647697712969323441, −4.11246691187800336651435789394, −2.95600226613732462711913907315, −2.28626078453958104937122590742, −1.04371521834324393162423492411,
1.04371521834324393162423492411, 2.28626078453958104937122590742, 2.95600226613732462711913907315, 4.11246691187800336651435789394, 4.89403119134647697712969323441, 5.87922716924499820150219554383, 6.52903438750615800839683333880, 7.48985068612798017467480657882, 8.173820523511472007915594724913, 8.944380108876022801650734435709
