| L(s) = 1 | + 0.816·5-s + 6.23·7-s + 7.41·11-s − 0.641i·13-s + 26.4·17-s + (−9.92 + 16.2i)19-s − 14.1·23-s − 24.3·25-s + 45.5i·29-s + 24.8i·31-s + 5.08·35-s + 15.7i·37-s + 23.7i·41-s + 47.2·43-s + 89.4·47-s + ⋯ |
| L(s) = 1 | + 0.163·5-s + 0.890·7-s + 0.674·11-s − 0.0493i·13-s + 1.55·17-s + (−0.522 + 0.852i)19-s − 0.615·23-s − 0.973·25-s + 1.57i·29-s + 0.801i·31-s + 0.145·35-s + 0.424i·37-s + 0.579i·41-s + 1.09·43-s + 1.90·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.458369765\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.458369765\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (9.92 - 16.2i)T \) |
| good | 5 | \( 1 - 0.816T + 25T^{2} \) |
| 7 | \( 1 - 6.23T + 49T^{2} \) |
| 11 | \( 1 - 7.41T + 121T^{2} \) |
| 13 | \( 1 + 0.641iT - 169T^{2} \) |
| 17 | \( 1 - 26.4T + 289T^{2} \) |
| 23 | \( 1 + 14.1T + 529T^{2} \) |
| 29 | \( 1 - 45.5iT - 841T^{2} \) |
| 31 | \( 1 - 24.8iT - 961T^{2} \) |
| 37 | \( 1 - 15.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 23.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 47.2T + 1.84e3T^{2} \) |
| 47 | \( 1 - 89.4T + 2.20e3T^{2} \) |
| 53 | \( 1 - 16.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 4.08iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 94.1T + 3.72e3T^{2} \) |
| 67 | \( 1 + 39.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 124. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 68.5T + 5.32e3T^{2} \) |
| 79 | \( 1 - 34.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 11.0T + 6.88e3T^{2} \) |
| 89 | \( 1 + 115. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 150. iT - 9.40e3T^{2} \) |
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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
−8.787052303782369332331481542136, −7.86902020501984274342502475941, −7.52082665070108710051209561687, −6.34455537431112524120874784986, −5.72465901091226343767100601177, −4.89729653314118281423969433709, −3.99316506267300882174765891623, −3.17826793479316641594602948490, −1.84762973310885932705657432909, −1.16371427233354070099920784023,
0.61024926365169890897044397871, 1.73325387764575567434211440554, 2.62561373896143661507945381997, 3.94669691694726960682940591447, 4.41102694411685117755689371455, 5.65956997191607063698861245175, 5.96174969700171893311864512550, 7.18891847837252388977951892407, 7.76173184797978609678025738170, 8.452058581633468905064379847933
