L(s) = 1 | + 2-s + 3·3-s − 4-s + 3·6-s + 3·7-s − 3·8-s + 6·9-s − 3·12-s − 4·13-s + 3·14-s − 16-s + 6·18-s + 4·19-s + 9·21-s + 8·23-s − 9·24-s − 4·26-s + 9·27-s − 3·28-s + 6·29-s − 2·31-s + 5·32-s − 6·36-s + 8·37-s + 4·38-s − 12·39-s − 5·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.73·3-s − 1/2·4-s + 1.22·6-s + 1.13·7-s − 1.06·8-s + 2·9-s − 0.866·12-s − 1.10·13-s + 0.801·14-s − 1/4·16-s + 1.41·18-s + 0.917·19-s + 1.96·21-s + 1.66·23-s − 1.83·24-s − 0.784·26-s + 1.73·27-s − 0.566·28-s + 1.11·29-s − 0.359·31-s + 0.883·32-s − 36-s + 1.31·37-s + 0.648·38-s − 1.92·39-s − 0.780·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.786671403\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.786671403\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−8.663893318244552167432000896179, −8.035964290055602491479098590864, −7.49408846715745525371895517734, −6.58718015938146113227777708713, −5.06691418009171028404488687631, −4.90884719144649574396593226669, −3.94051296020725456660139683008, −3.07436318446874405532232208556, −2.48136426259893877337842331261, −1.23569465875378898148634088028,
1.23569465875378898148634088028, 2.48136426259893877337842331261, 3.07436318446874405532232208556, 3.94051296020725456660139683008, 4.90884719144649574396593226669, 5.06691418009171028404488687631, 6.58718015938146113227777708713, 7.49408846715745525371895517734, 8.035964290055602491479098590864, 8.663893318244552167432000896179