L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 2·7-s − 8-s + 9-s + 12-s + 2·13-s − 2·14-s + 16-s − 17-s − 18-s − 4·19-s + 2·21-s − 6·23-s − 24-s − 5·25-s − 2·26-s + 27-s + 2·28-s − 10·31-s − 32-s + 34-s + 36-s + 8·37-s + 4·38-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.288·12-s + 0.554·13-s − 0.534·14-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 0.917·19-s + 0.436·21-s − 1.25·23-s − 0.204·24-s − 25-s − 0.392·26-s + 0.192·27-s + 0.377·28-s − 1.79·31-s − 0.176·32-s + 0.171·34-s + 1/6·36-s + 1.31·37-s + 0.648·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9286883281\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9286883281\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 17 | \( 1 + T \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.98628648985407066778492063059, −12.78638396593025333125074877118, −11.50655297040865569951307887714, −10.57689980236595016687233880100, −9.361746587069330588975575945669, −8.367508603382134314146293072792, −7.50641789037179894281138679970, −5.96956680213085492365415051586, −4.05372696693836488551925909574, −2.04126125287275460099535053657,
2.04126125287275460099535053657, 4.05372696693836488551925909574, 5.96956680213085492365415051586, 7.50641789037179894281138679970, 8.367508603382134314146293072792, 9.361746587069330588975575945669, 10.57689980236595016687233880100, 11.50655297040865569951307887714, 12.78638396593025333125074877118, 13.98628648985407066778492063059