| L(s) = 1 | + 3-s + 4·5-s − 2·9-s − 4·11-s + 5·13-s + 4·15-s + 2·17-s − 6·19-s + 23-s + 11·25-s − 5·27-s + 29-s + 9·31-s − 4·33-s − 4·37-s + 5·39-s − 3·41-s + 8·43-s − 8·45-s + 5·47-s + 2·51-s + 6·53-s − 16·55-s − 6·57-s + 4·59-s + 10·61-s + 20·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.78·5-s − 2/3·9-s − 1.20·11-s + 1.38·13-s + 1.03·15-s + 0.485·17-s − 1.37·19-s + 0.208·23-s + 11/5·25-s − 0.962·27-s + 0.185·29-s + 1.61·31-s − 0.696·33-s − 0.657·37-s + 0.800·39-s − 0.468·41-s + 1.21·43-s − 1.19·45-s + 0.729·47-s + 0.280·51-s + 0.824·53-s − 2.15·55-s − 0.794·57-s + 0.520·59-s + 1.28·61-s + 2.48·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.586904451\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.586904451\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 5 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 - 15 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−7.983329735683789641983119011839, −6.89898755225757394965457493437, −6.22427944100976515039703438465, −5.72071112885284431386619399514, −5.22616054505237962027078038483, −4.21062077000502854538751641371, −3.18159269469016319567712008976, −2.52176941377862984313607384670, −2.00206170859400532723065141572, −0.892674431167844766508019701449,
0.892674431167844766508019701449, 2.00206170859400532723065141572, 2.52176941377862984313607384670, 3.18159269469016319567712008976, 4.21062077000502854538751641371, 5.22616054505237962027078038483, 5.72071112885284431386619399514, 6.22427944100976515039703438465, 6.89898755225757394965457493437, 7.983329735683789641983119011839
