| L(s) = 1 | + 2-s − 1.08·3-s + 4-s − 0.588·5-s − 1.08·6-s − 3.89·7-s + 8-s − 1.81·9-s − 0.588·10-s − 1.57·11-s − 1.08·12-s − 2.79·13-s − 3.89·14-s + 0.639·15-s + 16-s − 4.34·17-s − 1.81·18-s − 8.24·19-s − 0.588·20-s + 4.22·21-s − 1.57·22-s + 23-s − 1.08·24-s − 4.65·25-s − 2.79·26-s + 5.23·27-s − 3.89·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.627·3-s + 0.5·4-s − 0.263·5-s − 0.443·6-s − 1.47·7-s + 0.353·8-s − 0.606·9-s − 0.186·10-s − 0.476·11-s − 0.313·12-s − 0.773·13-s − 1.03·14-s + 0.165·15-s + 0.250·16-s − 1.05·17-s − 0.428·18-s − 1.89·19-s − 0.131·20-s + 0.922·21-s − 0.336·22-s + 0.208·23-s − 0.221·24-s − 0.930·25-s − 0.547·26-s + 1.00·27-s − 0.735·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5389126377\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5389126377\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 - T \) |
| good | 3 | \( 1 + 1.08T + 3T^{2} \) |
| 5 | \( 1 + 0.588T + 5T^{2} \) |
| 7 | \( 1 + 3.89T + 7T^{2} \) |
| 11 | \( 1 + 1.57T + 11T^{2} \) |
| 13 | \( 1 + 2.79T + 13T^{2} \) |
| 17 | \( 1 + 4.34T + 17T^{2} \) |
| 19 | \( 1 + 8.24T + 19T^{2} \) |
| 29 | \( 1 - 2.87T + 29T^{2} \) |
| 31 | \( 1 + 4.33T + 31T^{2} \) |
| 37 | \( 1 - 0.877T + 37T^{2} \) |
| 41 | \( 1 - 0.656T + 41T^{2} \) |
| 43 | \( 1 - 8.44T + 43T^{2} \) |
| 47 | \( 1 + 7.04T + 47T^{2} \) |
| 53 | \( 1 - 11.4T + 53T^{2} \) |
| 59 | \( 1 - 3.46T + 59T^{2} \) |
| 61 | \( 1 - 1.74T + 61T^{2} \) |
| 67 | \( 1 - 2.58T + 67T^{2} \) |
| 71 | \( 1 - 12.9T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 + 9.17T + 79T^{2} \) |
| 83 | \( 1 + 17.2T + 83T^{2} \) |
| 89 | \( 1 - 1.26T + 89T^{2} \) |
| 97 | \( 1 + 1.72T + 97T^{2} \) |
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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
−8.018085007577954019387510542504, −6.91940662759208593226999208149, −6.65216421631068404063288978271, −5.90690388725772352261366039656, −5.32830895721702899499474465061, −4.38284083384635619587256515839, −3.80642944444531358247164034868, −2.73732016153474351440828902393, −2.26939008318507996059066217558, −0.32477314155498965333346653604,
0.32477314155498965333346653604, 2.26939008318507996059066217558, 2.73732016153474351440828902393, 3.80642944444531358247164034868, 4.38284083384635619587256515839, 5.32830895721702899499474465061, 5.90690388725772352261366039656, 6.65216421631068404063288978271, 6.91940662759208593226999208149, 8.018085007577954019387510542504
