| L(s) = 1 | + 5-s − 2.51i·11-s + 2.11i·13-s + 4.49·17-s + 5.65i·19-s − 7.98i·23-s + 25-s − 4.97i·29-s − 7.08i·31-s − 9.35·37-s − 6.23·41-s + 9.47·43-s − 10.4·47-s − 7.48i·53-s − 2.51i·55-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.757i·11-s + 0.585i·13-s + 1.09·17-s + 1.29i·19-s − 1.66i·23-s + 0.200·25-s − 0.923i·29-s − 1.27i·31-s − 1.53·37-s − 0.973·41-s + 1.44·43-s − 1.52·47-s − 1.02i·53-s − 0.338i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0980 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0980 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.669581381\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.669581381\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 2.51iT - 11T^{2} \) |
| 13 | \( 1 - 2.11iT - 13T^{2} \) |
| 17 | \( 1 - 4.49T + 17T^{2} \) |
| 19 | \( 1 - 5.65iT - 19T^{2} \) |
| 23 | \( 1 + 7.98iT - 23T^{2} \) |
| 29 | \( 1 + 4.97iT - 29T^{2} \) |
| 31 | \( 1 + 7.08iT - 31T^{2} \) |
| 37 | \( 1 + 9.35T + 37T^{2} \) |
| 41 | \( 1 + 6.23T + 41T^{2} \) |
| 43 | \( 1 - 9.47T + 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 + 7.48iT - 53T^{2} \) |
| 59 | \( 1 - 0.376T + 59T^{2} \) |
| 61 | \( 1 + 9.71iT - 61T^{2} \) |
| 67 | \( 1 + 3.75T + 67T^{2} \) |
| 71 | \( 1 - 9.01iT - 71T^{2} \) |
| 73 | \( 1 - 4.92iT - 73T^{2} \) |
| 79 | \( 1 + 15.9T + 79T^{2} \) |
| 83 | \( 1 - 2.29T + 83T^{2} \) |
| 89 | \( 1 - 17.7T + 89T^{2} \) |
| 97 | \( 1 + 4.14iT - 97T^{2} \) |
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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
−7.67711754060569651387406813745, −6.75736573627159569440700116683, −6.14692361793094985633225608553, −5.64703570579967479722785232577, −4.82954179190864250699383209934, −3.95992537719739497481946197686, −3.31196040977736328587177472122, −2.34243145991796667282614597963, −1.54341827445878254896276243763, −0.38513551597243584590912305500,
1.14711968842315383609180436946, 1.86814205128254046374971502097, 3.03813038735789803355326708034, 3.42711601055069754113767400482, 4.64647221736126018350192212279, 5.20153475529629905166909396610, 5.72675827911163208038444639632, 6.68987010749286050588762632088, 7.27697356804969789030588174312, 7.74691819419522329386714548184
