L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 7-s − 8-s − 2·9-s + 12-s − 13-s − 14-s + 16-s + 2·18-s − 4·19-s + 21-s + 23-s − 24-s + 26-s − 5·27-s + 28-s − 29-s + 2·31-s − 32-s − 2·36-s − 4·37-s + 4·38-s − 39-s + 3·41-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 − 2/3·9-s + 0.288·12-s − 0.277·13-s − 0.267·14-s + 1/4·16-s + 0.471·18-s − 0.917·19-s + 0.218·21-s + 0.208·23-s − 0.204·24-s + 0.196·26-s − 0.962·27-s + 0.188·28-s − 0.185·29-s + 0.359·31-s − 0.176·32-s − 1/3·36-s − 0.657·37-s + 0.648·38-s − 0.160·39-s + 0.468·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 233450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 233450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9535761691\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9535761691\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−12.98533932493009, −12.43699351934614, −11.80123492496855, −11.51632414183893, −10.96267754579749, −10.67515737238224, −9.867973899664705, −9.771059990137369, −8.989231854392356, −8.739589386354517, −8.242946680485698, −7.882037527663688, −7.433555968413456, −6.695507614154390, −6.437693696452675, −5.758338559118784, −5.212628725810997, −4.702829280827480, −4.023952761639889, −3.397771870464486, −2.944543051606289, −2.261139305660755, −1.916276091325925, −1.172420269193980, −0.2928414630383825,
0.2928414630383825, 1.172420269193980, 1.916276091325925, 2.261139305660755, 2.944543051606289, 3.397771870464486, 4.023952761639889, 4.702829280827480, 5.212628725810997, 5.758338559118784, 6.437693696452675, 6.695507614154390, 7.433555968413456, 7.882037527663688, 8.242946680485698, 8.739589386354517, 8.989231854392356, 9.771059990137369, 9.867973899664705, 10.67515737238224, 10.96267754579749, 11.51632414183893, 11.80123492496855, 12.43699351934614, 12.98533932493009