| L(s) = 1 | + 1.52·2-s − 3-s + 0.330·4-s − 0.362·5-s − 1.52·6-s − 2.54·8-s + 9-s − 0.552·10-s − 1.14·11-s − 0.330·12-s + 3.55·13-s + 0.362·15-s − 4.55·16-s + 4.81·17-s + 1.52·18-s − 4.61·19-s − 0.119·20-s − 1.75·22-s − 23-s + 2.54·24-s − 4.86·25-s + 5.43·26-s − 27-s + 1.29·29-s + 0.552·30-s + 9.28·31-s − 1.84·32-s + ⋯ |
| L(s) = 1 | + 1.07·2-s − 0.577·3-s + 0.165·4-s − 0.161·5-s − 0.623·6-s − 0.901·8-s + 0.333·9-s − 0.174·10-s − 0.346·11-s − 0.0953·12-s + 0.986·13-s + 0.0935·15-s − 1.13·16-s + 1.16·17-s + 0.359·18-s − 1.05·19-s − 0.0267·20-s − 0.373·22-s − 0.208·23-s + 0.520·24-s − 0.973·25-s + 1.06·26-s − 0.192·27-s + 0.239·29-s + 0.100·30-s + 1.66·31-s − 0.326·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.107525597\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.107525597\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - 1.52T + 2T^{2} \) |
| 5 | \( 1 + 0.362T + 5T^{2} \) |
| 11 | \( 1 + 1.14T + 11T^{2} \) |
| 13 | \( 1 - 3.55T + 13T^{2} \) |
| 17 | \( 1 - 4.81T + 17T^{2} \) |
| 19 | \( 1 + 4.61T + 19T^{2} \) |
| 29 | \( 1 - 1.29T + 29T^{2} \) |
| 31 | \( 1 - 9.28T + 31T^{2} \) |
| 37 | \( 1 + 9.08T + 37T^{2} \) |
| 41 | \( 1 - 6.09T + 41T^{2} \) |
| 43 | \( 1 - 6.31T + 43T^{2} \) |
| 47 | \( 1 - 0.0368T + 47T^{2} \) |
| 53 | \( 1 - 8.82T + 53T^{2} \) |
| 59 | \( 1 - 3.94T + 59T^{2} \) |
| 61 | \( 1 + 2.77T + 61T^{2} \) |
| 67 | \( 1 - 11.7T + 67T^{2} \) |
| 71 | \( 1 + 9.56T + 71T^{2} \) |
| 73 | \( 1 + 0.157T + 73T^{2} \) |
| 79 | \( 1 - 8.32T + 79T^{2} \) |
| 83 | \( 1 - 16.3T + 83T^{2} \) |
| 89 | \( 1 - 1.02T + 89T^{2} \) |
| 97 | \( 1 - 7.41T + 97T^{2} \) |
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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
−8.500844466524964960625083710027, −7.86851648928592700494792787024, −6.79891137560863942820308760430, −6.06036242055081341029376947049, −5.62732415380579423561652149485, −4.75880784867699697073310797571, −4.02761997090267600960412804491, −3.38335139177811212016693406609, −2.24985829218949329905430065741, −0.75302430383553227464502494061,
0.75302430383553227464502494061, 2.24985829218949329905430065741, 3.38335139177811212016693406609, 4.02761997090267600960412804491, 4.75880784867699697073310797571, 5.62732415380579423561652149485, 6.06036242055081341029376947049, 6.79891137560863942820308760430, 7.86851648928592700494792787024, 8.500844466524964960625083710027
