L(s) = 1 | + 2-s − 3-s + 4-s + 2·5-s − 6-s − 4·7-s + 8-s + 9-s + 2·10-s − 11-s − 12-s − 6·13-s − 4·14-s − 2·15-s + 16-s + 2·17-s + 18-s + 4·19-s + 2·20-s + 4·21-s − 22-s + 4·23-s − 24-s − 25-s − 6·26-s − 27-s − 4·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.632·10-s − 0.301·11-s − 0.288·12-s − 1.66·13-s − 1.06·14-s − 0.516·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.917·19-s + 0.447·20-s + 0.872·21-s − 0.213·22-s + 0.834·23-s − 0.204·24-s − 1/5·25-s − 1.17·26-s − 0.192·27-s − 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 66 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.102192530\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.102192530\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−14.77427121841588439759624877733, −13.55277453788152412700214007654, −12.76064311034148466780729958842, −11.85600244708104500189303464848, −10.17335307448534281740885152086, −9.626249861423627809345281403043, −7.24765397306950965920878295914, −6.15685276163708486295790955262, −5.03220048416044134767486652990, −2.90847219285821389032814298970,
2.90847219285821389032814298970, 5.03220048416044134767486652990, 6.15685276163708486295790955262, 7.24765397306950965920878295914, 9.626249861423627809345281403043, 10.17335307448534281740885152086, 11.85600244708104500189303464848, 12.76064311034148466780729958842, 13.55277453788152412700214007654, 14.77427121841588439759624877733