L(s) = 1 | − 2·2-s + 2·4-s − 5-s + 3·7-s + 2·10-s + 11-s − 13-s − 6·14-s − 4·16-s + 17-s − 2·19-s − 2·20-s − 2·22-s + 3·23-s + 25-s + 2·26-s + 6·28-s + 2·29-s − 6·31-s + 8·32-s − 2·34-s − 3·35-s + 11·37-s + 4·38-s + 5·41-s + 4·43-s + 2·44-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 4-s − 0.447·5-s + 1.13·7-s + 0.632·10-s + 0.301·11-s − 0.277·13-s − 1.60·14-s − 16-s + 0.242·17-s − 0.458·19-s − 0.447·20-s − 0.426·22-s + 0.625·23-s + 1/5·25-s + 0.392·26-s + 1.13·28-s + 0.371·29-s − 1.07·31-s + 1.41·32-s − 0.342·34-s − 0.507·35-s + 1.80·37-s + 0.648·38-s + 0.780·41-s + 0.609·43-s + 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7407218622\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7407218622\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
good | 2 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−10.87030732385610107944198759200, −9.599414851193643593710511267052, −8.985183316745324235009539175161, −7.996671602095315844360321530012, −7.64083008607609627592622911960, −6.54697003120115263269660356276, −5.09278902217219401825829455179, −4.09635384552794966459642537511, −2.30897204101866244390058598775, −0.980772475336589212801861022003,
0.980772475336589212801861022003, 2.30897204101866244390058598775, 4.09635384552794966459642537511, 5.09278902217219401825829455179, 6.54697003120115263269660356276, 7.64083008607609627592622911960, 7.996671602095315844360321530012, 8.985183316745324235009539175161, 9.599414851193643593710511267052, 10.87030732385610107944198759200