L(s) = 1 | + 2-s + 3-s − 4-s − 2·5-s + 6-s + 7-s − 3·8-s + 9-s − 2·10-s − 11-s − 12-s − 6·13-s + 14-s − 2·15-s − 16-s + 2·17-s + 18-s + 2·20-s + 21-s − 22-s − 3·24-s − 25-s − 6·26-s + 27-s − 28-s + 2·29-s − 2·30-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 − 1.06·8-s + 1/3·9-s − 0.632·10-s − 0.301·11-s − 0.288·12-s − 1.66·13-s + 0.267·14-s − 0.516·15-s − 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.447·20-s + 0.218·21-s − 0.213·22-s − 0.612·24-s − 1/5·25-s − 1.17·26-s + 0.192·27-s − 0.188·28-s + 0.371·29-s − 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 83391 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83391 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 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 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 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
−14.37057914884708, −14.06366267630270, −13.55519785747596, −12.87643485468884, −12.51890455191232, −12.15065152982820, −11.64779985043995, −11.16025405737205, −10.36058630794687, −9.835334615784587, −9.548872396507056, −8.768103014782007, −8.376942332333106, −7.882044659935690, −7.366943205074850, −6.942193908180665, −6.159177601176456, −5.316150615592865, −5.026444844056619, −4.574011503419736, −3.904491351667717, −3.373726018039744, −2.972048188477216, −2.151083353293982, −1.432398174032259, 0, 0,
1.432398174032259, 2.151083353293982, 2.972048188477216, 3.373726018039744, 3.904491351667717, 4.574011503419736, 5.026444844056619, 5.316150615592865, 6.159177601176456, 6.942193908180665, 7.366943205074850, 7.882044659935690, 8.376942332333106, 8.768103014782007, 9.548872396507056, 9.835334615784587, 10.36058630794687, 11.16025405737205, 11.64779985043995, 12.15065152982820, 12.51890455191232, 12.87643485468884, 13.55519785747596, 14.06366267630270, 14.37057914884708