L(s) = 1 | − 0.805·2-s + (−0.972 − 1.43i)3-s − 1.35·4-s − i·5-s + (0.783 + 1.15i)6-s + i·7-s + 2.69·8-s + (−1.10 + 2.78i)9-s + 0.805i·10-s + (−0.889 + 3.19i)11-s + (1.31 + 1.93i)12-s − 0.669i·13-s − 0.805i·14-s + (−1.43 + 0.972i)15-s + 0.530·16-s + 5.06·17-s + ⋯ |
L(s) = 1 | − 0.569·2-s + (−0.561 − 0.827i)3-s − 0.675·4-s − 0.447i·5-s + (0.319 + 0.471i)6-s + 0.377i·7-s + 0.954·8-s + (−0.369 + 0.929i)9-s + 0.254i·10-s + (−0.268 + 0.963i)11-s + (0.379 + 0.559i)12-s − 0.185i·13-s − 0.215i·14-s + (−0.370 + 0.251i)15-s + 0.132·16-s + 1.22·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.947 + 0.318i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.947 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4075958934\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4075958934\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.972 + 1.43i)T \) |
| 5 | \( 1 + iT \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 + (0.889 - 3.19i)T \) |
good | 2 | \( 1 + 0.805T + 2T^{2} \) |
| 13 | \( 1 + 0.669iT - 13T^{2} \) |
| 17 | \( 1 - 5.06T + 17T^{2} \) |
| 19 | \( 1 + 8.50iT - 19T^{2} \) |
| 23 | \( 1 + 6.77iT - 23T^{2} \) |
| 29 | \( 1 + 2.44T + 29T^{2} \) |
| 31 | \( 1 - 5.73T + 31T^{2} \) |
| 37 | \( 1 + 4.26T + 37T^{2} \) |
| 41 | \( 1 + 10.5T + 41T^{2} \) |
| 43 | \( 1 - 1.79iT - 43T^{2} \) |
| 47 | \( 1 - 7.04iT - 47T^{2} \) |
| 53 | \( 1 - 9.79iT - 53T^{2} \) |
| 59 | \( 1 - 11.6iT - 59T^{2} \) |
| 61 | \( 1 + 7.58iT - 61T^{2} \) |
| 67 | \( 1 + 9.11T + 67T^{2} \) |
| 71 | \( 1 + 11.8iT - 71T^{2} \) |
| 73 | \( 1 + 12.3iT - 73T^{2} \) |
| 79 | \( 1 + 11.9iT - 79T^{2} \) |
| 83 | \( 1 - 6.36T + 83T^{2} \) |
| 89 | \( 1 + 8.93iT - 89T^{2} \) |
| 97 | \( 1 + 9.76T + 97T^{2} \) |
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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
−9.299213142743503294664616351884, −8.605306121830590267663171883156, −7.81024383110695936301154781233, −7.14899471304668853907813523935, −6.10594962012545619938429962529, −4.96194509542408601933004676457, −4.66437685515926232459881154824, −2.80406332596196995609730338792, −1.49161490713109562953911516734, −0.27674357672433780550087008789,
1.28135740929285687083581250344, 3.50211328286538189232374906249, 3.79355492791907739285606913104, 5.25509022302838914407681286523, 5.67101748566717955667979770111, 6.88316122996760743986291982713, 8.005200791500199439258652197306, 8.484141070122585012282735097091, 9.730131728119974430620000471285, 9.992311718567017316336759542779