| L(s) = 1 | + 2-s + 3-s + 4-s − 2·5-s + 6-s − 7-s + 8-s + 9-s − 2·10-s + 12-s + 6·13-s − 14-s − 2·15-s + 16-s + 6·17-s + 18-s − 2·20-s − 21-s − 23-s + 24-s − 25-s + 6·26-s + 27-s − 28-s + 10·29-s − 2·30-s + 5·31-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.632·10-s + 0.288·12-s + 1.66·13-s − 0.267·14-s − 0.516·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.447·20-s − 0.218·21-s − 0.208·23-s + 0.204·24-s − 1/5·25-s + 1.17·26-s + 0.192·27-s − 0.188·28-s + 1.85·29-s − 0.365·30-s + 0.898·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.088000819\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.088000819\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 11 T + p T^{2} \) |
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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
−13.67908641143926, −13.13590579793308, −12.69833968259872, −12.14369052997889, −11.74891171200959, −11.37522456356458, −10.66946504926329, −10.26950432856309, −9.757837686957733, −9.151986219939873, −8.413290410518608, −8.126536784983099, −7.790818632437571, −7.101443500668197, −6.466446930401194, −6.109859249726764, −5.578827757291150, −4.658231562705687, −4.406060531537120, −3.614487529784569, −3.357242296994250, −2.920188987435511, −2.065725455871503, −1.190091583643426, −0.7304387822007663,
0.7304387822007663, 1.190091583643426, 2.065725455871503, 2.920188987435511, 3.357242296994250, 3.614487529784569, 4.406060531537120, 4.658231562705687, 5.578827757291150, 6.109859249726764, 6.466446930401194, 7.101443500668197, 7.790818632437571, 8.126536784983099, 8.413290410518608, 9.151986219939873, 9.757837686957733, 10.26950432856309, 10.66946504926329, 11.37522456356458, 11.74891171200959, 12.14369052997889, 12.69833968259872, 13.13590579793308, 13.67908641143926
