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 − 2·13-s − 14-s − 2·15-s − 16-s − 2·17-s + 18-s − 2·20-s + 21-s + 22-s − 8·23-s + 3·24-s − 25-s − 2·26-s − 27-s + 28-s − 29-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 − 0.554·13-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 − 1.66·23-s + 0.612·24-s − 1/5·25-s − 0.392·26-s − 0.192·27-s + 0.188·28-s − 0.185·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6699 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6699 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.606066106\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.606066106\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 29 | \( 1 + T \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−7.959360366488323939859153674708, −6.99360553048504259763972147727, −6.31661926206400859990989164830, −5.75108350630500043474138182762, −5.30681786756950550485095809524, −4.31756274516913988573923423411, −3.93556838926677774859501798274, −2.76167171878264865451346326257, −1.97350481302744602365385622380, −0.58734554816749409283376644854,
0.58734554816749409283376644854, 1.97350481302744602365385622380, 2.76167171878264865451346326257, 3.93556838926677774859501798274, 4.31756274516913988573923423411, 5.30681786756950550485095809524, 5.75108350630500043474138182762, 6.31661926206400859990989164830, 6.99360553048504259763972147727, 7.959360366488323939859153674708