| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 4·7-s − 8-s + 9-s − 10-s − 12-s + 2·13-s + 4·14-s − 15-s + 16-s − 18-s − 19-s + 20-s + 4·21-s + 24-s + 25-s − 2·26-s − 27-s − 4·28-s − 10·29-s + 30-s − 32-s − 4·35-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s + 0.554·13-s + 1.06·14-s − 0.258·15-s + 1/4·16-s − 0.235·18-s − 0.229·19-s + 0.223·20-s + 0.872·21-s + 0.204·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s − 0.755·28-s − 1.85·29-s + 0.182·30-s − 0.176·32-s − 0.676·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2779619598\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2779619598\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 17 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−13.07760800565717, −12.73424312050097, −12.42478296929216, −11.70854409729458, −11.25476425060108, −10.81484814562791, −10.36202519519259, −9.775689405488116, −9.560390806870495, −9.108525710217192, −8.526146304348884, −7.973239933889316, −7.307654531966027, −6.866213002005035, −6.340369052959496, −6.168012453656268, −5.365148578277732, −5.152593516748953, −3.963004940316578, −3.752534158844769, −3.047001639808733, −2.414489113101007, −1.737948868282514, −1.094186106223988, −0.1931046924143775,
0.1931046924143775, 1.094186106223988, 1.737948868282514, 2.414489113101007, 3.047001639808733, 3.752534158844769, 3.963004940316578, 5.152593516748953, 5.365148578277732, 6.168012453656268, 6.340369052959496, 6.866213002005035, 7.307654531966027, 7.973239933889316, 8.526146304348884, 9.108525710217192, 9.560390806870495, 9.775689405488116, 10.36202519519259, 10.81484814562791, 11.25476425060108, 11.70854409729458, 12.42478296929216, 12.73424312050097, 13.07760800565717
