| L(s) = 1 | − 2.70i·2-s − 5.34·4-s + (−2.17 − 0.539i)5-s − i·7-s + 9.04i·8-s + (−1.46 + 5.87i)10-s − 2·11-s + 0.921i·13-s − 2.70·14-s + 13.8·16-s + 1.07i·17-s − 3.07·19-s + (11.5 + 2.87i)20-s + 5.41i·22-s − 2.34i·23-s + ⋯ |
| L(s) = 1 | − 1.91i·2-s − 2.67·4-s + (−0.970 − 0.241i)5-s − 0.377i·7-s + 3.19i·8-s + (−0.461 + 1.85i)10-s − 0.603·11-s + 0.255i·13-s − 0.724·14-s + 3.45·16-s + 0.261i·17-s − 0.706·19-s + (2.59 + 0.643i)20-s + 1.15i·22-s − 0.487i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.241 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.241 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.224995 + 0.175933i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.224995 + 0.175933i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (2.17 + 0.539i)T \) |
| 7 | \( 1 + iT \) |
| good | 2 | \( 1 + 2.70iT - 2T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 0.921iT - 13T^{2} \) |
| 17 | \( 1 - 1.07iT - 17T^{2} \) |
| 19 | \( 1 + 3.07T + 19T^{2} \) |
| 23 | \( 1 + 2.34iT - 23T^{2} \) |
| 29 | \( 1 + 6.68T + 29T^{2} \) |
| 31 | \( 1 + 7.75T + 31T^{2} \) |
| 37 | \( 1 + 10.8iT - 37T^{2} \) |
| 41 | \( 1 + 6.49T + 41T^{2} \) |
| 43 | \( 1 + 6.52iT - 43T^{2} \) |
| 47 | \( 1 - 4.68iT - 47T^{2} \) |
| 53 | \( 1 + 3.75iT - 53T^{2} \) |
| 59 | \( 1 - 10.5T + 59T^{2} \) |
| 61 | \( 1 + 4.15T + 61T^{2} \) |
| 67 | \( 1 + 4.68iT - 67T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 - 7.07iT - 73T^{2} \) |
| 79 | \( 1 + 6.15T + 79T^{2} \) |
| 83 | \( 1 + 6.83iT - 83T^{2} \) |
| 89 | \( 1 - 8.34T + 89T^{2} \) |
| 97 | \( 1 - 8.43iT - 97T^{2} \) |
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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
−10.96759238313537829291146569725, −10.41475350496298204558637951709, −9.224179736239260135051470834294, −8.483148036344066353086286906321, −7.40818868282286244524281058301, −5.35057922288324676362073761334, −4.22532333419140520618556305315, −3.51484980101028708031524218240, −2.02512130961847861631874876543, −0.20603202974116156237081984781,
3.51075496372994254386916156301, 4.72086934782513074781873583635, 5.63717914403602623176442666667, 6.75173872392127389999505442314, 7.57408143214044730958608428362, 8.269811187421069973926153330370, 9.106930211602389459010011300356, 10.25608241744790500812627319686, 11.55323335783874440874123282602, 12.78208218864548396349541568475
