| L(s) = 1 | + (0.125 + 1.99i)2-s + (−3.96 + 0.5i)4-s − 7i·7-s + (−1.49 − 7.85i)8-s + 10.7i·11-s + (13.9 − 0.876i)14-s + (15.5 − 3.96i)16-s + (−21.4 + 1.34i)22-s + 42.6·23-s + 25·25-s + (3.5 + 27.7i)28-s + 53.1·29-s + (9.86 + 30.4i)32-s + 38i·37-s − 63.4·43-s + (−5.36 − 42.5i)44-s + ⋯ |
| L(s) = 1 | + (0.0626 + 0.998i)2-s + (−0.992 + 0.125i)4-s − i·7-s + (−0.186 − 0.982i)8-s + 0.974i·11-s + (0.998 − 0.0626i)14-s + (0.968 − 0.248i)16-s + (−0.972 + 0.0610i)22-s + 1.85·23-s + 25-s + (0.125 + 0.992i)28-s + 1.83·29-s + (0.308 + 0.951i)32-s + 1.02i·37-s − 1.47·43-s + (−0.121 − 0.967i)44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.414 - 0.910i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.414 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.621274219\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.621274219\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.125 - 1.99i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 7iT \) |
| good | 5 | \( 1 - 25T^{2} \) |
| 11 | \( 1 - 10.7iT - 121T^{2} \) |
| 13 | \( 1 + 169T^{2} \) |
| 17 | \( 1 + 289T^{2} \) |
| 19 | \( 1 - 361T^{2} \) |
| 23 | \( 1 - 42.6T + 529T^{2} \) |
| 29 | \( 1 - 53.1T + 841T^{2} \) |
| 31 | \( 1 + 961T^{2} \) |
| 37 | \( 1 - 38iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 1.68e3T^{2} \) |
| 43 | \( 1 + 63.4T + 1.84e3T^{2} \) |
| 47 | \( 1 - 2.20e3T^{2} \) |
| 53 | \( 1 - 70.5T + 2.80e3T^{2} \) |
| 59 | \( 1 + 3.48e3T^{2} \) |
| 61 | \( 1 + 3.72e3T^{2} \) |
| 67 | \( 1 - 118T + 4.48e3T^{2} \) |
| 71 | \( 1 + 20.7T + 5.04e3T^{2} \) |
| 73 | \( 1 - 5.32e3T^{2} \) |
| 79 | \( 1 + 126. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 6.88e3T^{2} \) |
| 89 | \( 1 + 7.92e3T^{2} \) |
| 97 | \( 1 - 9.40e3T^{2} \) |
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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
−10.56685727200437230295480826685, −9.948154052771946931318846469025, −8.919634313674043350688083045983, −8.046673634607313498424146826953, −6.96707411568113434009709781473, −6.68685838172408353149913518769, −5.06827287432507811670398965640, −4.50874855952039356633631739517, −3.18893808517264751575406978622, −0.983942319042473080295851855140,
0.931010661372727483022640520731, 2.54098042987703457804786781806, 3.34514903291790191768094031661, 4.80925022828712682061856115813, 5.57112837791180294093644134668, 6.77229225033460987477388043637, 8.420414693819105731507236536478, 8.739079910965622589664791777511, 9.723604449646956153841148008401, 10.75733637271215319176269388697
