| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.39641 + 0.863034i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.39641 + 0.863034i\) |
| \(L(\frac{3}{2})\) |
|
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
−10.96204283325738727631663142423, −10.36437055045198716504439160744, −9.596895731312918570686994084414, −8.822716182437411563588507555940, −7.54555479629257307603786610140, −6.51874874837716799735560030318, −5.56528834141212959111599732244, −4.54527580989995364796753291808, −3.21614344233243892520096554711, −1.88186128453870152472191643390,
1.19025606975213521752961924701, 2.46255277453660539042908429911, 4.17671272069787922677323321393, 5.27540683747586444306474300080, 6.34779806164796797682815056701, 7.20401396368338010417614047920, 8.173451757540625524449103432871, 9.239515366937217046208034029278, 10.05228025991007960734073145114, 10.82363709223638425886916429788
