| L(s) = 1 | − 1.41i·3-s + 5-s − 3.46·7-s + 0.999·9-s + (1.73 − 2.82i)11-s + 2.44i·13-s − 1.41i·15-s − 7.34i·17-s − 3.46·19-s + 4.89i·21-s − 1.41i·23-s + 25-s − 5.65i·27-s − 4.89i·29-s + (−4.00 − 2.44i)33-s + ⋯ |
| L(s) = 1 | − 0.816i·3-s + 0.447·5-s − 1.30·7-s + 0.333·9-s + (0.522 − 0.852i)11-s + 0.679i·13-s − 0.365i·15-s − 1.78i·17-s − 0.794·19-s + 1.06i·21-s − 0.294i·23-s + 0.200·25-s − 1.08i·27-s − 0.909i·29-s + (−0.696 − 0.426i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.522 + 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.616200 - 1.09990i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.616200 - 1.09990i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + (-1.73 + 2.82i)T \) |
| good | 3 | \( 1 + 1.41iT - 3T^{2} \) |
| 7 | \( 1 + 3.46T + 7T^{2} \) |
| 13 | \( 1 - 2.44iT - 13T^{2} \) |
| 17 | \( 1 + 7.34iT - 17T^{2} \) |
| 19 | \( 1 + 3.46T + 19T^{2} \) |
| 23 | \( 1 + 1.41iT - 23T^{2} \) |
| 29 | \( 1 + 4.89iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 3.46T + 43T^{2} \) |
| 47 | \( 1 + 7.07iT - 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 2.82iT - 59T^{2} \) |
| 61 | \( 1 - 9.79iT - 61T^{2} \) |
| 67 | \( 1 + 12.7iT - 67T^{2} \) |
| 71 | \( 1 - 5.65iT - 71T^{2} \) |
| 73 | \( 1 - 7.34iT - 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 - 12T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−9.647423119341164715191268001577, −9.209393782920618786661781930436, −8.142170643561120431072452849905, −6.89391106960445124531507372891, −6.67739698358866920171842339879, −5.76260173485134442525644345628, −4.41811681320383772132919493717, −3.21773274376224227164490406373, −2.13350737286571655270967702605, −0.60079907078416439953282864151,
1.74441559480447174465210878098, 3.28675815834908145579279477773, 4.01530886996188396723436066872, 5.06352411353058574993002668635, 6.21336740968476408882762922841, 6.74343540152821515679999388310, 7.965243262780232382810463773548, 9.107916519078049493887943088530, 9.609880259575423246770459343356, 10.41059712656660763821943979003
