L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 7-s − 8-s + 9-s − 11-s + 12-s + 2·13-s − 14-s + 16-s + 6·17-s − 18-s − 4·19-s + 21-s + 22-s − 24-s − 2·26-s + 27-s + 28-s − 2·29-s − 8·31-s − 32-s − 33-s − 6·34-s + 36-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.301·11-s + 0.288·12-s + 0.554·13-s − 0.267·14-s + 1/4·16-s + 1.45·17-s − 0.235·18-s − 0.917·19-s + 0.218·21-s + 0.213·22-s − 0.204·24-s − 0.392·26-s + 0.192·27-s + 0.188·28-s − 0.371·29-s − 1.43·31-s − 0.176·32-s − 0.174·33-s − 1.02·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
good | 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 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 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−16.76228025180740, −16.17661123904265, −15.65960128680590, −14.93333725003412, −14.59871839624118, −14.00514130048966, −13.26597118989986, −12.69727012718933, −12.13780458601285, −11.40397544845374, −10.80142975263865, −10.29429112990051, −9.697967261513995, −9.032958305024325, −8.427064063279043, −8.040879445227861, −7.321531379981400, −6.818546703326838, −5.831331873720643, −5.366137402151700, −4.341311598243866, −3.547775147137137, −2.935490157448273, −1.892639945365149, −1.359549432601987, 0,
1.359549432601987, 1.892639945365149, 2.935490157448273, 3.547775147137137, 4.341311598243866, 5.366137402151700, 5.831331873720643, 6.818546703326838, 7.321531379981400, 8.040879445227861, 8.427064063279043, 9.032958305024325, 9.697967261513995, 10.29429112990051, 10.80142975263865, 11.40397544845374, 12.13780458601285, 12.69727012718933, 13.26597118989986, 14.00514130048966, 14.59871839624118, 14.93333725003412, 15.65960128680590, 16.17661123904265, 16.76228025180740