L(s) = 1 | − 2-s + 2.93·3-s + 4-s − 2.47·5-s − 2.93·6-s + 4.98·7-s − 8-s + 5.59·9-s + 2.47·10-s − 11-s + 2.93·12-s + 4.23·13-s − 4.98·14-s − 7.26·15-s + 16-s + 0.103·17-s − 5.59·18-s − 1.47·19-s − 2.47·20-s + 14.6·21-s + 22-s + 7.68·23-s − 2.93·24-s + 1.14·25-s − 4.23·26-s + 7.61·27-s + 4.98·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.69·3-s + 0.5·4-s − 1.10·5-s − 1.19·6-s + 1.88·7-s − 0.353·8-s + 1.86·9-s + 0.783·10-s − 0.301·11-s + 0.846·12-s + 1.17·13-s − 1.33·14-s − 1.87·15-s + 0.250·16-s + 0.0251·17-s − 1.31·18-s − 0.338·19-s − 0.554·20-s + 3.18·21-s + 0.213·22-s + 1.60·23-s − 0.598·24-s + 0.229·25-s − 0.829·26-s + 1.46·27-s + 0.941·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.996718674\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.996718674\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 197 | \( 1 - T \) |
good | 3 | \( 1 - 2.93T + 3T^{2} \) |
| 5 | \( 1 + 2.47T + 5T^{2} \) |
| 7 | \( 1 - 4.98T + 7T^{2} \) |
| 13 | \( 1 - 4.23T + 13T^{2} \) |
| 17 | \( 1 - 0.103T + 17T^{2} \) |
| 19 | \( 1 + 1.47T + 19T^{2} \) |
| 23 | \( 1 - 7.68T + 23T^{2} \) |
| 29 | \( 1 - 2.31T + 29T^{2} \) |
| 31 | \( 1 - 5.41T + 31T^{2} \) |
| 37 | \( 1 + 1.88T + 37T^{2} \) |
| 41 | \( 1 + 6.79T + 41T^{2} \) |
| 43 | \( 1 + 5.78T + 43T^{2} \) |
| 47 | \( 1 + 3.21T + 47T^{2} \) |
| 53 | \( 1 - 1.01T + 53T^{2} \) |
| 59 | \( 1 + 3.45T + 59T^{2} \) |
| 61 | \( 1 - 2.90T + 61T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 - 0.898T + 71T^{2} \) |
| 73 | \( 1 - 13.7T + 73T^{2} \) |
| 79 | \( 1 + 0.156T + 79T^{2} \) |
| 83 | \( 1 - 5.32T + 83T^{2} \) |
| 89 | \( 1 - 9.25T + 89T^{2} \) |
| 97 | \( 1 - 9.71T + 97T^{2} \) |
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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
−8.294656763058153889951509481181, −8.056688058841910931336511144323, −7.37267774251831505912681247239, −6.63711994379289692873898098478, −5.15919785212745920876259175916, −4.42760190245964517375480531786, −3.62959124371499339378586402772, −2.87690191314510953157544058773, −1.85675116237902164338009561166, −1.10896158832323310043992321974,
1.10896158832323310043992321974, 1.85675116237902164338009561166, 2.87690191314510953157544058773, 3.62959124371499339378586402772, 4.42760190245964517375480531786, 5.15919785212745920876259175916, 6.63711994379289692873898098478, 7.37267774251831505912681247239, 8.056688058841910931336511144323, 8.294656763058153889951509481181