| L(s) = 1 | − 3-s + 5-s + 2·7-s + 9-s − 2·11-s + 2·13-s − 15-s − 8·17-s + 4·19-s − 2·21-s + 23-s + 25-s − 27-s + 2·29-s + 8·31-s + 2·33-s + 2·35-s + 8·37-s − 2·39-s + 2·41-s + 8·43-s + 45-s + 8·47-s − 3·49-s + 8·51-s − 6·53-s − 2·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s − 0.603·11-s + 0.554·13-s − 0.258·15-s − 1.94·17-s + 0.917·19-s − 0.436·21-s + 0.208·23-s + 1/5·25-s − 0.192·27-s + 0.371·29-s + 1.43·31-s + 0.348·33-s + 0.338·35-s + 1.31·37-s − 0.320·39-s + 0.312·41-s + 1.21·43-s + 0.149·45-s + 1.16·47-s − 3/7·49-s + 1.12·51-s − 0.824·53-s − 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.737371596\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.737371596\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 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
−8.893896280820227288738761898527, −7.994796476745490899886913228438, −7.33234531907817580566969416299, −6.31002267459628598380929857125, −5.88807712399053566637896240878, −4.71185119126099264262610017109, −4.50194373326221280659789712692, −2.98587665012523684593495161016, −2.03366966442391978363614613380, −0.864257625040359920482110279546,
0.864257625040359920482110279546, 2.03366966442391978363614613380, 2.98587665012523684593495161016, 4.50194373326221280659789712692, 4.71185119126099264262610017109, 5.88807712399053566637896240878, 6.31002267459628598380929857125, 7.33234531907817580566969416299, 7.994796476745490899886913228438, 8.893896280820227288738761898527
