| L(s) = 1 | + i·3-s + (2 − i)5-s + i·7-s + 2·9-s + 11-s + (1 + 2i)15-s − i·17-s + 19-s − 21-s + (3 − 4i)25-s + 5i·27-s + 29-s + 31-s + i·33-s + (1 + 2i)35-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + (0.894 − 0.447i)5-s + 0.377i·7-s + 0.666·9-s + 0.301·11-s + (0.258 + 0.516i)15-s − 0.242i·17-s + 0.229·19-s − 0.218·21-s + (0.600 − 0.800i)25-s + 0.962i·27-s + 0.185·29-s + 0.179·31-s + 0.174i·33-s + (0.169 + 0.338i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.94650 + 0.459508i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.94650 + 0.459508i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-2 + i)T \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 - iT - 3T^{2} \) |
| 7 | \( 1 - iT - 7T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + iT - 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - T + 29T^{2} \) |
| 31 | \( 1 - T + 31T^{2} \) |
| 37 | \( 1 + iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 6iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 - 9iT - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 7T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 5T + 71T^{2} \) |
| 73 | \( 1 - 14iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 16iT - 83T^{2} \) |
| 89 | \( 1 - 7T + 89T^{2} \) |
| 97 | \( 1 - 16iT - 97T^{2} \) |
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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
−10.13898238215407571663734279985, −9.313045457277553068622508299338, −8.897743975286703712645999652806, −7.68351829663461662789449949694, −6.65626184671590531540661971855, −5.72286456632019030961288228297, −4.90208226681841181061947190100, −4.01156032124327073947183754904, −2.64608944348735857152053018683, −1.35150114447800051738599593194,
1.23609497013526401921280739250, 2.30114734216782872578266309561, 3.60471629920815387070579509162, 4.79098303196483030742787910605, 5.93158406856047361028501662416, 6.70993892708165864018464839749, 7.32222619783932134211396301169, 8.323655504051748825063387532869, 9.403301977539821374141066299398, 10.07564368925042555799865877887
