| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 7-s − 8-s + 9-s − 10-s + 3·11-s − 12-s − 5·13-s + 14-s − 15-s + 16-s − 8·17-s − 18-s + 5·19-s + 20-s + 21-s − 3·22-s − 3·23-s + 24-s + 25-s + 5·26-s − 27-s − 28-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 + 0.904·11-s − 0.288·12-s − 1.38·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s − 1.94·17-s − 0.235·18-s + 1.14·19-s + 0.223·20-s + 0.218·21-s − 0.639·22-s − 0.625·23-s + 0.204·24-s + 1/5·25-s + 0.980·26-s − 0.192·27-s − 0.188·28-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
| \(L(1)\) |
\(\approx\) |
\(1.302073425\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.302073425\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 11 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 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
−12.57832574937123, −12.15745883492933, −11.80409867938126, −11.25925605268138, −10.92662493434253, −10.37788676280237, −9.903694030566824, −9.473944021518409, −9.050371554490786, −8.931190322961071, −7.977459710758943, −7.509282541652723, −7.071960435116306, −6.706983746615640, −6.247512811993391, −5.601724587960967, −5.342855145988311, −4.520725854948325, −4.139614915001945, −3.538989541543585, −2.669155825501402, −2.237258879353802, −1.824177050581954, −0.8819575217631743, −0.4380422989589679,
0.4380422989589679, 0.8819575217631743, 1.824177050581954, 2.237258879353802, 2.669155825501402, 3.538989541543585, 4.139614915001945, 4.520725854948325, 5.342855145988311, 5.601724587960967, 6.247512811993391, 6.706983746615640, 7.071960435116306, 7.509282541652723, 7.977459710758943, 8.931190322961071, 9.050371554490786, 9.473944021518409, 9.903694030566824, 10.37788676280237, 10.92662493434253, 11.25925605268138, 11.80409867938126, 12.15745883492933, 12.57832574937123
