| L(s) = 1 | − 2·5-s + 4.24·7-s − 3·9-s + 4.24·11-s − 13-s + 4·17-s + 4.24·19-s − 25-s − 2·29-s + 4.24·31-s − 8.48·35-s − 6·37-s + 2·41-s − 8.48·43-s + 6·45-s + 12.7·47-s + 10.9·49-s − 8·53-s − 8.48·55-s − 4.24·59-s − 12.7·63-s + 2·65-s + 4.24·67-s + 4.24·71-s + 6·73-s + 17.9·77-s − 8.48·79-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.60·7-s − 9-s + 1.27·11-s − 0.277·13-s + 0.970·17-s + 0.973·19-s − 0.200·25-s − 0.371·29-s + 0.762·31-s − 1.43·35-s − 0.986·37-s + 0.312·41-s − 1.29·43-s + 0.894·45-s + 1.85·47-s + 1.57·49-s − 1.09·53-s − 1.14·55-s − 0.552·59-s − 1.60·63-s + 0.248·65-s + 0.518·67-s + 0.503·71-s + 0.702·73-s + 2.05·77-s − 0.954·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.955261439\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.955261439\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 3T^{2} \) |
| 5 | \( 1 + 2T + 5T^{2} \) |
| 7 | \( 1 - 4.24T + 7T^{2} \) |
| 11 | \( 1 - 4.24T + 11T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 - 4.24T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 4.24T + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 8.48T + 43T^{2} \) |
| 47 | \( 1 - 12.7T + 47T^{2} \) |
| 53 | \( 1 + 8T + 53T^{2} \) |
| 59 | \( 1 + 4.24T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 4.24T + 67T^{2} \) |
| 71 | \( 1 - 4.24T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 8.48T + 79T^{2} \) |
| 83 | \( 1 - 12.7T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 18T + 97T^{2} \) |
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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
−8.541485438164292735418107891576, −7.81995041110292028237710070100, −7.46650868845705331512986323841, −6.37297320641009731219027408655, −5.43832227181315514769757423559, −4.84147234427561243175964105754, −3.91496065289375490933073148483, −3.21330476887886852338836372284, −1.90765950066416831346590094818, −0.876470278411069950884027055131,
0.876470278411069950884027055131, 1.90765950066416831346590094818, 3.21330476887886852338836372284, 3.91496065289375490933073148483, 4.84147234427561243175964105754, 5.43832227181315514769757423559, 6.37297320641009731219027408655, 7.46650868845705331512986323841, 7.81995041110292028237710070100, 8.541485438164292735418107891576
