| L(s) = 1 | + 3-s − 3·7-s − 2·9-s − 5·11-s − 5·17-s + 19-s − 3·21-s + 23-s − 5·27-s − 9·29-s + 4·31-s − 5·33-s − 3·37-s + 41-s + 3·43-s − 8·47-s + 2·49-s − 5·51-s − 10·53-s + 57-s − 3·59-s + 7·61-s + 6·63-s − 9·67-s + 69-s − 7·71-s + 10·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.13·7-s − 2/3·9-s − 1.50·11-s − 1.21·17-s + 0.229·19-s − 0.654·21-s + 0.208·23-s − 0.962·27-s − 1.67·29-s + 0.718·31-s − 0.870·33-s − 0.493·37-s + 0.156·41-s + 0.457·43-s − 1.16·47-s + 2/7·49-s − 0.700·51-s − 1.37·53-s + 0.132·57-s − 0.390·59-s + 0.896·61-s + 0.755·63-s − 1.09·67-s + 0.120·69-s − 0.830·71-s + 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−15.52789734188559, −15.03984863112107, −14.51699635358694, −13.77369280266016, −13.36914141597584, −13.04006852014239, −12.58218624083750, −11.82936586828046, −11.08828129958516, −10.87082857205989, −10.08740384452382, −9.515459868771393, −9.194954516437586, −8.459429674177775, −8.027743213435559, −7.424894724216196, −6.783415377372715, −6.187029937147108, −5.557535992530743, −5.036389466981956, −4.186136331765969, −3.476805178504849, −2.828339241429025, −2.533441016903323, −1.610729375238264, 0, 0,
1.610729375238264, 2.533441016903323, 2.828339241429025, 3.476805178504849, 4.186136331765969, 5.036389466981956, 5.557535992530743, 6.187029937147108, 6.783415377372715, 7.424894724216196, 8.027743213435559, 8.459429674177775, 9.194954516437586, 9.515459868771393, 10.08740384452382, 10.87082857205989, 11.08828129958516, 11.82936586828046, 12.58218624083750, 13.04006852014239, 13.36914141597584, 13.77369280266016, 14.51699635358694, 15.03984863112107, 15.52789734188559
