| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 7-s − 8-s + 9-s + 10-s + 6·11-s − 12-s − 3·13-s + 14-s + 15-s + 16-s + 6·17-s − 18-s − 20-s + 21-s − 6·22-s − 3·23-s + 24-s + 25-s + 3·26-s − 27-s − 28-s + 3·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.80·11-s − 0.288·12-s − 0.832·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s + 1.45·17-s − 0.235·18-s − 0.223·20-s + 0.218·21-s − 1.27·22-s − 0.625·23-s + 0.204·24-s + 1/5·25-s + 0.588·26-s − 0.192·27-s − 0.188·28-s + 0.557·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.823664300\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.823664300\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 37 | \( 1 \) |
| good | 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 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
−12.42575765878048, −12.22858571164871, −11.76963427949213, −11.48447509748460, −10.81996077268093, −10.42557179516140, −9.923524437630618, −9.475687624292172, −9.167886489346074, −8.671406781905686, −7.970935720648009, −7.559252711228600, −7.193592000940522, −6.635696748982504, −6.233716459150045, −5.774884953504671, −5.141810062074207, −4.626988800877777, −3.895083087523524, −3.532439193734054, −3.093837244717416, −2.020800759220697, −1.772263905498000, −0.7366560057376807, −0.6405617840692379,
0.6405617840692379, 0.7366560057376807, 1.772263905498000, 2.020800759220697, 3.093837244717416, 3.532439193734054, 3.895083087523524, 4.626988800877777, 5.141810062074207, 5.774884953504671, 6.233716459150045, 6.635696748982504, 7.193592000940522, 7.559252711228600, 7.970935720648009, 8.671406781905686, 9.167886489346074, 9.475687624292172, 9.923524437630618, 10.42557179516140, 10.81996077268093, 11.48447509748460, 11.76963427949213, 12.22858571164871, 12.42575765878048
