| L(s) = 1 | + (1 + 1.41i)3-s − i·5-s + 0.828i·7-s + (−1.00 + 2.82i)9-s − 0.828·11-s − 6.82·13-s + (1.41 − i)15-s + 4.82i·17-s + 6i·19-s + (−1.17 + 0.828i)21-s + 4.82·23-s − 25-s + (−5.00 + 1.41i)27-s + 6i·29-s + 2i·31-s + ⋯ |
| L(s) = 1 | + (0.577 + 0.816i)3-s − 0.447i·5-s + 0.313i·7-s + (−0.333 + 0.942i)9-s − 0.249·11-s − 1.89·13-s + (0.365 − 0.258i)15-s + 1.17i·17-s + 1.37i·19-s + (−0.255 + 0.180i)21-s + 1.00·23-s − 0.200·25-s + (−0.962 + 0.272i)27-s + 1.11i·29-s + 0.359i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.623102 + 1.20374i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.623102 + 1.20374i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1 - 1.41i)T \) |
| 5 | \( 1 + iT \) |
| good | 7 | \( 1 - 0.828iT - 7T^{2} \) |
| 11 | \( 1 + 0.828T + 11T^{2} \) |
| 13 | \( 1 + 6.82T + 13T^{2} \) |
| 17 | \( 1 - 4.82iT - 17T^{2} \) |
| 19 | \( 1 - 6iT - 19T^{2} \) |
| 23 | \( 1 - 4.82T + 23T^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 - 2iT - 31T^{2} \) |
| 37 | \( 1 + 1.17T + 37T^{2} \) |
| 41 | \( 1 - 1.65iT - 41T^{2} \) |
| 43 | \( 1 + 6.82iT - 43T^{2} \) |
| 47 | \( 1 - 8.82T + 47T^{2} \) |
| 53 | \( 1 - 3.65iT - 53T^{2} \) |
| 59 | \( 1 - 7.17T + 59T^{2} \) |
| 61 | \( 1 + 9.31T + 61T^{2} \) |
| 67 | \( 1 + 12.4iT - 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 - 6iT - 79T^{2} \) |
| 83 | \( 1 - 17.3T + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−10.31634410407111899985739548471, −9.420820900062018258890477373633, −8.780976908028003295142327559595, −7.956003334789838103123225040575, −7.16125397011257398292963863467, −5.69194332417270120411632559774, −5.03677756902174926644935968214, −4.11443753035682927731730057662, −3.02829383146172825813076352534, −1.91537324480910516310394356639,
0.55531404142086397403715491696, 2.43507371833463527432156924322, 2.85805945065386545787723181419, 4.38832025763114647239075010323, 5.37919165809038594767477739801, 6.69637394051890865558779125688, 7.27918622735936250058180809601, 7.73909224668552255437444309810, 9.031623997655702065616084320319, 9.543072496241233007726291639856
