| L(s) = 1 | − 1.36·3-s + 2.92·5-s + 7-s − 1.14·9-s + 0.360·11-s − 4.33·13-s − 3.97·15-s − 3.98·17-s + 19-s − 1.36·21-s + 5.24·23-s + 3.54·25-s + 5.64·27-s + 2.89·29-s + 0.582·31-s − 0.490·33-s + 2.92·35-s − 9.96·37-s + 5.90·39-s − 5.88·41-s − 3.12·43-s − 3.36·45-s − 6.44·47-s + 49-s + 5.42·51-s − 4.09·53-s + 1.05·55-s + ⋯ |
| L(s) = 1 | − 0.785·3-s + 1.30·5-s + 0.377·7-s − 0.383·9-s + 0.108·11-s − 1.20·13-s − 1.02·15-s − 0.966·17-s + 0.229·19-s − 0.296·21-s + 1.09·23-s + 0.709·25-s + 1.08·27-s + 0.536·29-s + 0.104·31-s − 0.0853·33-s + 0.494·35-s − 1.63·37-s + 0.945·39-s − 0.918·41-s − 0.476·43-s − 0.500·45-s − 0.940·47-s + 0.142·49-s + 0.759·51-s − 0.562·53-s + 0.142·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4256 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 1.36T + 3T^{2} \) |
| 5 | \( 1 - 2.92T + 5T^{2} \) |
| 11 | \( 1 - 0.360T + 11T^{2} \) |
| 13 | \( 1 + 4.33T + 13T^{2} \) |
| 17 | \( 1 + 3.98T + 17T^{2} \) |
| 23 | \( 1 - 5.24T + 23T^{2} \) |
| 29 | \( 1 - 2.89T + 29T^{2} \) |
| 31 | \( 1 - 0.582T + 31T^{2} \) |
| 37 | \( 1 + 9.96T + 37T^{2} \) |
| 41 | \( 1 + 5.88T + 41T^{2} \) |
| 43 | \( 1 + 3.12T + 43T^{2} \) |
| 47 | \( 1 + 6.44T + 47T^{2} \) |
| 53 | \( 1 + 4.09T + 53T^{2} \) |
| 59 | \( 1 + 9.30T + 59T^{2} \) |
| 61 | \( 1 + 4.51T + 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 + 1.59T + 71T^{2} \) |
| 73 | \( 1 - 3.12T + 73T^{2} \) |
| 79 | \( 1 + 5.95T + 79T^{2} \) |
| 83 | \( 1 - 10.4T + 83T^{2} \) |
| 89 | \( 1 + 17.0T + 89T^{2} \) |
| 97 | \( 1 - 12.4T + 97T^{2} \) |
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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
−8.101644565128747714107061093274, −6.92273448303553059697663976619, −6.62515261933044106316639474529, −5.72006611404618534890703619605, −5.06105234673582101913706694368, −4.71753252228727708964858036562, −3.19415179279491811932363995648, −2.33782342477724506444386470113, −1.45494857461975500701468006442, 0,
1.45494857461975500701468006442, 2.33782342477724506444386470113, 3.19415179279491811932363995648, 4.71753252228727708964858036562, 5.06105234673582101913706694368, 5.72006611404618534890703619605, 6.62515261933044106316639474529, 6.92273448303553059697663976619, 8.101644565128747714107061093274
