L(s) = 1 | − 2-s − 4-s + 3·8-s − 4·11-s − 13-s − 16-s + 2·17-s − 4·19-s + 4·22-s + 8·23-s + 26-s + 2·29-s − 8·31-s − 5·32-s − 2·34-s − 6·37-s + 4·38-s + 6·41-s + 4·43-s + 4·44-s − 8·46-s − 8·47-s − 7·49-s + 52-s + 6·53-s − 2·58-s + 12·59-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.06·8-s − 1.20·11-s − 0.277·13-s − 1/4·16-s + 0.485·17-s − 0.917·19-s + 0.852·22-s + 1.66·23-s + 0.196·26-s + 0.371·29-s − 1.43·31-s − 0.883·32-s − 0.342·34-s − 0.986·37-s + 0.648·38-s + 0.937·41-s + 0.609·43-s + 0.603·44-s − 1.17·46-s − 1.16·47-s − 49-s + 0.138·52-s + 0.824·53-s − 0.262·58-s + 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7612303932\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7612303932\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−8.728931928185248388257372058485, −8.136025378033966872311574564843, −7.43193712425490303306328835346, −6.75028913136328658806730701739, −5.44064201059248414897390242722, −5.04341051321205815078664291033, −4.05811586767484409743342375016, −3.01337399574887020794792151462, −1.90651535301238430158520971763, −0.59299171903779565408875923090,
0.59299171903779565408875923090, 1.90651535301238430158520971763, 3.01337399574887020794792151462, 4.05811586767484409743342375016, 5.04341051321205815078664291033, 5.44064201059248414897390242722, 6.75028913136328658806730701739, 7.43193712425490303306328835346, 8.136025378033966872311574564843, 8.728931928185248388257372058485