L(s) = 1 | − 2-s + 4-s + 4·5-s + 3·7-s − 8-s − 4·10-s − 2·11-s − 13-s − 3·14-s + 16-s − 3·17-s − 19-s + 4·20-s + 2·22-s + 23-s + 11·25-s + 26-s + 3·28-s + 5·29-s − 8·31-s − 32-s + 3·34-s + 12·35-s − 2·37-s + 38-s − 4·40-s + 8·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.78·5-s + 1.13·7-s − 0.353·8-s − 1.26·10-s − 0.603·11-s − 0.277·13-s − 0.801·14-s + 1/4·16-s − 0.727·17-s − 0.229·19-s + 0.894·20-s + 0.426·22-s + 0.208·23-s + 11/5·25-s + 0.196·26-s + 0.566·28-s + 0.928·29-s − 1.43·31-s − 0.176·32-s + 0.514·34-s + 2.02·35-s − 0.328·37-s + 0.162·38-s − 0.632·40-s + 1.24·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.360215538\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.360215538\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 15 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−11.07673504239234599196375456535, −10.60506565264478658567127520158, −9.607471665269029768409922323294, −8.900666531045489798996891080014, −7.892225159593170281658461026008, −6.73926084905849932341893089023, −5.70493789281341106083458944039, −4.80751556264464963514931464778, −2.55734550860524537659292391699, −1.62231881828892009952147969605,
1.62231881828892009952147969605, 2.55734550860524537659292391699, 4.80751556264464963514931464778, 5.70493789281341106083458944039, 6.73926084905849932341893089023, 7.892225159593170281658461026008, 8.900666531045489798996891080014, 9.607471665269029768409922323294, 10.60506565264478658567127520158, 11.07673504239234599196375456535