| L(s) = 1 | + 2·3-s + 5-s + 2·7-s + 9-s − 2·13-s + 2·15-s − 6·17-s + 4·19-s + 4·21-s + 6·23-s + 25-s − 4·27-s − 6·29-s − 4·31-s + 2·35-s − 2·37-s − 4·39-s + 6·41-s + 10·43-s + 45-s − 6·47-s − 3·49-s − 12·51-s + 6·53-s + 8·57-s − 12·59-s − 2·61-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s − 0.554·13-s + 0.516·15-s − 1.45·17-s + 0.917·19-s + 0.872·21-s + 1.25·23-s + 1/5·25-s − 0.769·27-s − 1.11·29-s − 0.718·31-s + 0.338·35-s − 0.328·37-s − 0.640·39-s + 0.937·41-s + 1.52·43-s + 0.149·45-s − 0.875·47-s − 3/7·49-s − 1.68·51-s + 0.824·53-s + 1.05·57-s − 1.56·59-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.997134815\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.997134815\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−11.48400710378319677524604221692, −10.75706369377722819329609640960, −9.332287203335424542049109475711, −9.039858624083400649706600435206, −7.87489459574230399033328236522, −7.09464940360169401329405651291, −5.59481538925271546647259599831, −4.45194901772205251160909130654, −3.01436611750632173441455711709, −1.91202694614300604029027352933,
1.91202694614300604029027352933, 3.01436611750632173441455711709, 4.45194901772205251160909130654, 5.59481538925271546647259599831, 7.09464940360169401329405651291, 7.87489459574230399033328236522, 9.039858624083400649706600435206, 9.332287203335424542049109475711, 10.75706369377722819329609640960, 11.48400710378319677524604221692
