| L(s) = 1 | − 1.80·3-s + 5-s + 1.80·7-s + 0.246·9-s − 1.55·11-s − 1.80·15-s + 0.911·17-s − 1.69·19-s − 3.24·21-s + 0.246·23-s + 25-s + 4.96·27-s − 5.29·29-s + 1.08·31-s + 2.80·33-s + 1.80·35-s − 3·37-s − 6.96·41-s + 2.26·43-s + 0.246·45-s − 8.52·47-s − 3.75·49-s − 1.64·51-s + 6.92·53-s − 1.55·55-s + 3.04·57-s + 14.7·59-s + ⋯ |
| L(s) = 1 | − 1.04·3-s + 0.447·5-s + 0.681·7-s + 0.0823·9-s − 0.468·11-s − 0.465·15-s + 0.221·17-s − 0.388·19-s − 0.708·21-s + 0.0514·23-s + 0.200·25-s + 0.954·27-s − 0.983·29-s + 0.195·31-s + 0.487·33-s + 0.304·35-s − 0.493·37-s − 1.08·41-s + 0.345·43-s + 0.0368·45-s − 1.24·47-s − 0.536·49-s − 0.230·51-s + 0.951·53-s − 0.209·55-s + 0.403·57-s + 1.91·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 1.80T + 3T^{2} \) |
| 7 | \( 1 - 1.80T + 7T^{2} \) |
| 11 | \( 1 + 1.55T + 11T^{2} \) |
| 17 | \( 1 - 0.911T + 17T^{2} \) |
| 19 | \( 1 + 1.69T + 19T^{2} \) |
| 23 | \( 1 - 0.246T + 23T^{2} \) |
| 29 | \( 1 + 5.29T + 29T^{2} \) |
| 31 | \( 1 - 1.08T + 31T^{2} \) |
| 37 | \( 1 + 3T + 37T^{2} \) |
| 41 | \( 1 + 6.96T + 41T^{2} \) |
| 43 | \( 1 - 2.26T + 43T^{2} \) |
| 47 | \( 1 + 8.52T + 47T^{2} \) |
| 53 | \( 1 - 6.92T + 53T^{2} \) |
| 59 | \( 1 - 14.7T + 59T^{2} \) |
| 61 | \( 1 + 9.55T + 61T^{2} \) |
| 67 | \( 1 + 5.21T + 67T^{2} \) |
| 71 | \( 1 - 5.74T + 71T^{2} \) |
| 73 | \( 1 + 14.5T + 73T^{2} \) |
| 79 | \( 1 - 7.75T + 79T^{2} \) |
| 83 | \( 1 + 4.94T + 83T^{2} \) |
| 89 | \( 1 - 11.9T + 89T^{2} \) |
| 97 | \( 1 - 9.17T + 97T^{2} \) |
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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
−8.269026556334588702184712863689, −7.42143503371270736411940878808, −6.60119951860150371260790089181, −5.88545569162775808917498006668, −5.22478052761812439407099729601, −4.71922467943821447371810909860, −3.54843799702503170132519463516, −2.39428561416103794863585868480, −1.36839303892555848008040630391, 0,
1.36839303892555848008040630391, 2.39428561416103794863585868480, 3.54843799702503170132519463516, 4.71922467943821447371810909860, 5.22478052761812439407099729601, 5.88545569162775808917498006668, 6.60119951860150371260790089181, 7.42143503371270736411940878808, 8.269026556334588702184712863689
