L(s) = 1 | − 2-s + 4-s + 5-s − 7-s − 8-s − 10-s − 4·11-s + 6·13-s + 14-s + 16-s + 4·17-s + 6·19-s + 20-s + 4·22-s + 25-s − 6·26-s − 28-s − 6·29-s − 4·31-s − 32-s − 4·34-s − 35-s + 8·37-s − 6·38-s − 40-s + 10·41-s − 2·43-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s − 0.353·8-s − 0.316·10-s − 1.20·11-s + 1.66·13-s + 0.267·14-s + 1/4·16-s + 0.970·17-s + 1.37·19-s + 0.223·20-s + 0.852·22-s + 1/5·25-s − 1.17·26-s − 0.188·28-s − 1.11·29-s − 0.718·31-s − 0.176·32-s − 0.685·34-s − 0.169·35-s + 1.31·37-s − 0.973·38-s − 0.158·40-s + 1.56·41-s − 0.304·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.158516587\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.158516587\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−10.56743428972484569359598566056, −9.651318147210564421578847637382, −9.011176212020501533758656307057, −7.926382084447849379211800748461, −7.31560604921550912264416152995, −5.95452960277078004817932527329, −5.51252842562545301276873448163, −3.73367739437301890472425811296, −2.65030952280904272753344625283, −1.09983097292649461808831986293,
1.09983097292649461808831986293, 2.65030952280904272753344625283, 3.73367739437301890472425811296, 5.51252842562545301276873448163, 5.95452960277078004817932527329, 7.31560604921550912264416152995, 7.926382084447849379211800748461, 9.011176212020501533758656307057, 9.651318147210564421578847637382, 10.56743428972484569359598566056