| L(s) = 1 | + 2-s − 2.23·3-s + 4-s + 1.50·5-s − 2.23·6-s − 1.71·7-s + 8-s + 2.01·9-s + 1.50·10-s + 1.21·11-s − 2.23·12-s − 0.896·13-s − 1.71·14-s − 3.36·15-s + 16-s − 2.01·17-s + 2.01·18-s + 3.44·19-s + 1.50·20-s + 3.84·21-s + 1.21·22-s + 8.91·23-s − 2.23·24-s − 2.73·25-s − 0.896·26-s + 2.21·27-s − 1.71·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.29·3-s + 0.5·4-s + 0.672·5-s − 0.913·6-s − 0.648·7-s + 0.353·8-s + 0.670·9-s + 0.475·10-s + 0.367·11-s − 0.646·12-s − 0.248·13-s − 0.458·14-s − 0.869·15-s + 0.250·16-s − 0.488·17-s + 0.474·18-s + 0.789·19-s + 0.336·20-s + 0.838·21-s + 0.259·22-s + 1.85·23-s − 0.456·24-s − 0.547·25-s − 0.175·26-s + 0.425·27-s − 0.324·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1502 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1502 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.783568428\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.783568428\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 751 | \( 1 + T \) |
| good | 3 | \( 1 + 2.23T + 3T^{2} \) |
| 5 | \( 1 - 1.50T + 5T^{2} \) |
| 7 | \( 1 + 1.71T + 7T^{2} \) |
| 11 | \( 1 - 1.21T + 11T^{2} \) |
| 13 | \( 1 + 0.896T + 13T^{2} \) |
| 17 | \( 1 + 2.01T + 17T^{2} \) |
| 19 | \( 1 - 3.44T + 19T^{2} \) |
| 23 | \( 1 - 8.91T + 23T^{2} \) |
| 29 | \( 1 - 0.676T + 29T^{2} \) |
| 31 | \( 1 - 1.18T + 31T^{2} \) |
| 37 | \( 1 + 4.11T + 37T^{2} \) |
| 41 | \( 1 - 5.39T + 41T^{2} \) |
| 43 | \( 1 - 1.94T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 - 1.99T + 53T^{2} \) |
| 59 | \( 1 + 4.28T + 59T^{2} \) |
| 61 | \( 1 - 3.55T + 61T^{2} \) |
| 67 | \( 1 - 13.6T + 67T^{2} \) |
| 71 | \( 1 - 0.411T + 71T^{2} \) |
| 73 | \( 1 - 5.19T + 73T^{2} \) |
| 79 | \( 1 + 6.16T + 79T^{2} \) |
| 83 | \( 1 - 11.5T + 83T^{2} \) |
| 89 | \( 1 - 15.3T + 89T^{2} \) |
| 97 | \( 1 - 8.52T + 97T^{2} \) |
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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
−9.624966614016846457316531741399, −8.910439030465173141268485692937, −7.44454400284611326889231865110, −6.70572784706192266964841230328, −6.11921375879706886849917687814, −5.37749524823179757176441994718, −4.75573138334822770553078797831, −3.55605769721797006469872201906, −2.43800935928039434340944373859, −0.929377123818511144739636879174,
0.929377123818511144739636879174, 2.43800935928039434340944373859, 3.55605769721797006469872201906, 4.75573138334822770553078797831, 5.37749524823179757176441994718, 6.11921375879706886849917687814, 6.70572784706192266964841230328, 7.44454400284611326889231865110, 8.910439030465173141268485692937, 9.624966614016846457316531741399
