| L(s) = 1 | + 2-s + 4-s + 5-s + 8-s + 10-s − 3·11-s + 5·13-s + 16-s + 5·19-s + 20-s − 3·22-s + 9·23-s + 25-s + 5·26-s − 10·31-s + 32-s − 37-s + 5·38-s + 40-s − 9·41-s + 8·43-s − 3·44-s + 9·46-s − 3·47-s + 50-s + 5·52-s + 3·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.353·8-s + 0.316·10-s − 0.904·11-s + 1.38·13-s + 1/4·16-s + 1.14·19-s + 0.223·20-s − 0.639·22-s + 1.87·23-s + 1/5·25-s + 0.980·26-s − 1.79·31-s + 0.176·32-s − 0.164·37-s + 0.811·38-s + 0.158·40-s − 1.40·41-s + 1.21·43-s − 0.452·44-s + 1.32·46-s − 0.437·47-s + 0.141·50-s + 0.693·52-s + 0.412·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.666480980\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.666480980\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−8.341444344167051154643369075413, −7.43305614706180418880288993656, −6.88669076415307490226672747307, −5.97114650284778311607794821095, −5.36516961588533318774124083733, −4.84031150360678981783742055036, −3.60282309940883986324694901049, −3.14793615569999932765104141824, −2.05704464759869960308258648437, −1.01929647973571985185229926989,
1.01929647973571985185229926989, 2.05704464759869960308258648437, 3.14793615569999932765104141824, 3.60282309940883986324694901049, 4.84031150360678981783742055036, 5.36516961588533318774124083733, 5.97114650284778311607794821095, 6.88669076415307490226672747307, 7.43305614706180418880288993656, 8.341444344167051154643369075413
