| L(s) = 1 | + 2.91·3-s − 4.30·5-s − 1.78·7-s + 5.47·9-s − 0.138·11-s − 3.10·13-s − 12.5·15-s + 0.510·17-s + 4.60·19-s − 5.20·21-s + 0.162·23-s + 13.4·25-s + 7.19·27-s + 7.39·29-s + 6.83·31-s − 0.403·33-s + 7.68·35-s + 1.26·37-s − 9.04·39-s − 7.71·41-s − 5.85·43-s − 23.5·45-s − 9.92·47-s − 3.80·49-s + 1.48·51-s − 6.61·53-s + 0.596·55-s + ⋯ |
| L(s) = 1 | + 1.68·3-s − 1.92·5-s − 0.675·7-s + 1.82·9-s − 0.0418·11-s − 0.861·13-s − 3.23·15-s + 0.123·17-s + 1.05·19-s − 1.13·21-s + 0.0337·23-s + 2.69·25-s + 1.38·27-s + 1.37·29-s + 1.22·31-s − 0.0703·33-s + 1.29·35-s + 0.207·37-s − 1.44·39-s − 1.20·41-s − 0.892·43-s − 3.50·45-s − 1.44·47-s − 0.543·49-s + 0.208·51-s − 0.909·53-s + 0.0804·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 251 | \( 1 + T \) |
| good | 3 | \( 1 - 2.91T + 3T^{2} \) |
| 5 | \( 1 + 4.30T + 5T^{2} \) |
| 7 | \( 1 + 1.78T + 7T^{2} \) |
| 11 | \( 1 + 0.138T + 11T^{2} \) |
| 13 | \( 1 + 3.10T + 13T^{2} \) |
| 17 | \( 1 - 0.510T + 17T^{2} \) |
| 19 | \( 1 - 4.60T + 19T^{2} \) |
| 23 | \( 1 - 0.162T + 23T^{2} \) |
| 29 | \( 1 - 7.39T + 29T^{2} \) |
| 31 | \( 1 - 6.83T + 31T^{2} \) |
| 37 | \( 1 - 1.26T + 37T^{2} \) |
| 41 | \( 1 + 7.71T + 41T^{2} \) |
| 43 | \( 1 + 5.85T + 43T^{2} \) |
| 47 | \( 1 + 9.92T + 47T^{2} \) |
| 53 | \( 1 + 6.61T + 53T^{2} \) |
| 59 | \( 1 + 6.68T + 59T^{2} \) |
| 61 | \( 1 - 3.96T + 61T^{2} \) |
| 67 | \( 1 + 0.974T + 67T^{2} \) |
| 71 | \( 1 - 1.00T + 71T^{2} \) |
| 73 | \( 1 + 13.2T + 73T^{2} \) |
| 79 | \( 1 + 2.20T + 79T^{2} \) |
| 83 | \( 1 - 14.3T + 83T^{2} \) |
| 89 | \( 1 - 1.75T + 89T^{2} \) |
| 97 | \( 1 + 4.15T + 97T^{2} \) |
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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
−7.76121667892707750765032813817, −7.00742767814853356689683581362, −6.56109909358906251317949588460, −4.93999753487647058683257965466, −4.52076373549550738776301143882, −3.59195494977657320127512018804, −3.14912920533030496659563872638, −2.71289955427265540820437505977, −1.29638869606445004570161430305, 0,
1.29638869606445004570161430305, 2.71289955427265540820437505977, 3.14912920533030496659563872638, 3.59195494977657320127512018804, 4.52076373549550738776301143882, 4.93999753487647058683257965466, 6.56109909358906251317949588460, 7.00742767814853356689683581362, 7.76121667892707750765032813817
