L(s) = 1 | + 2-s + 0.732·3-s + 4-s + 0.732·6-s − 7-s + 8-s − 2.46·9-s + 11-s + 0.732·12-s − 5.46·13-s − 14-s + 16-s + 3.46·17-s − 2.46·18-s + 0.732·19-s − 0.732·21-s + 22-s − 4.73·23-s + 0.732·24-s − 5.46·26-s − 4·27-s − 28-s − 1.26·29-s − 4.92·31-s + 32-s + 0.732·33-s + 3.46·34-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.422·3-s + 0.5·4-s + 0.298·6-s − 0.377·7-s + 0.353·8-s − 0.821·9-s + 0.301·11-s + 0.211·12-s − 1.51·13-s − 0.267·14-s + 0.250·16-s + 0.840·17-s − 0.580·18-s + 0.167·19-s − 0.159·21-s + 0.213·22-s − 0.986·23-s + 0.149·24-s − 1.07·26-s − 0.769·27-s − 0.188·28-s − 0.235·29-s − 0.885·31-s + 0.176·32-s + 0.127·33-s + 0.594·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
good | 3 | \( 1 - 0.732T + 3T^{2} \) |
| 13 | \( 1 + 5.46T + 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 - 0.732T + 19T^{2} \) |
| 23 | \( 1 + 4.73T + 23T^{2} \) |
| 29 | \( 1 + 1.26T + 29T^{2} \) |
| 31 | \( 1 + 4.92T + 31T^{2} \) |
| 37 | \( 1 + 6.73T + 37T^{2} \) |
| 41 | \( 1 + 1.26T + 41T^{2} \) |
| 43 | \( 1 + 8.92T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 1.26T + 53T^{2} \) |
| 59 | \( 1 - 13.8T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 2.92T + 67T^{2} \) |
| 71 | \( 1 - 2.53T + 71T^{2} \) |
| 73 | \( 1 + 4.53T + 73T^{2} \) |
| 79 | \( 1 - 3.26T + 79T^{2} \) |
| 83 | \( 1 + 16.3T + 83T^{2} \) |
| 89 | \( 1 + 8.53T + 89T^{2} \) |
| 97 | \( 1 + 16.1T + 97T^{2} \) |
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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
−8.066233460339454264682989348086, −7.27620223359216700117805352198, −6.69946625258572726051352453113, −5.58778594917576167237089416214, −5.31847341040317025652784444149, −4.16124959646750260153692002425, −3.40497355091257219311234723298, −2.68497131524621931696567812514, −1.79377042301192777583866116513, 0,
1.79377042301192777583866116513, 2.68497131524621931696567812514, 3.40497355091257219311234723298, 4.16124959646750260153692002425, 5.31847341040317025652784444149, 5.58778594917576167237089416214, 6.69946625258572726051352453113, 7.27620223359216700117805352198, 8.066233460339454264682989348086