| L(s) = 1 | + 3-s + 3.56·5-s + 9-s − 1.56·11-s + 0.438·13-s + 3.56·15-s + 17-s + 4.68·19-s + 2.43·23-s + 7.68·25-s + 27-s − 8.24·29-s − 3.12·31-s − 1.56·33-s − 5.12·37-s + 0.438·39-s − 3.56·41-s − 4.68·43-s + 3.56·45-s + 11.1·47-s − 7·49-s + 51-s + 12.2·53-s − 5.56·55-s + 4.68·57-s − 7.12·59-s + 9.12·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.59·5-s + 0.333·9-s − 0.470·11-s + 0.121·13-s + 0.919·15-s + 0.242·17-s + 1.07·19-s + 0.508·23-s + 1.53·25-s + 0.192·27-s − 1.53·29-s − 0.560·31-s − 0.271·33-s − 0.842·37-s + 0.0702·39-s − 0.556·41-s − 0.714·43-s + 0.530·45-s + 1.62·47-s − 49-s + 0.140·51-s + 1.68·53-s − 0.749·55-s + 0.620·57-s − 0.927·59-s + 1.16·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.463929358\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.463929358\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - 3.56T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 1.56T + 11T^{2} \) |
| 13 | \( 1 - 0.438T + 13T^{2} \) |
| 19 | \( 1 - 4.68T + 19T^{2} \) |
| 23 | \( 1 - 2.43T + 23T^{2} \) |
| 29 | \( 1 + 8.24T + 29T^{2} \) |
| 31 | \( 1 + 3.12T + 31T^{2} \) |
| 37 | \( 1 + 5.12T + 37T^{2} \) |
| 41 | \( 1 + 3.56T + 41T^{2} \) |
| 43 | \( 1 + 4.68T + 43T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 + 7.12T + 59T^{2} \) |
| 61 | \( 1 - 9.12T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 - 6.24T + 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 - 9.36T + 79T^{2} \) |
| 83 | \( 1 - 0.876T + 83T^{2} \) |
| 89 | \( 1 + 1.12T + 89T^{2} \) |
| 97 | \( 1 + 2.87T + 97T^{2} \) |
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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
−10.09279835718349523545198282752, −9.392889917208226312016326903477, −8.800559452458532457002293782550, −7.64267525713506852230626064025, −6.82490296717831663376459765954, −5.67154765581505412951760428848, −5.16425264635517540706805771418, −3.60293958039380029029989342158, −2.52372285764627412038394937469, −1.51361677777597230861383744778,
1.51361677777597230861383744778, 2.52372285764627412038394937469, 3.60293958039380029029989342158, 5.16425264635517540706805771418, 5.67154765581505412951760428848, 6.82490296717831663376459765954, 7.64267525713506852230626064025, 8.800559452458532457002293782550, 9.392889917208226312016326903477, 10.09279835718349523545198282752
