L(s) = 1 | − 1.58i·3-s + (−2.09 − 0.787i)5-s + 2.84i·7-s + 0.493·9-s + 1.98·11-s + 4.69i·13-s + (−1.24 + 3.31i)15-s − 3.16i·17-s + 6.16·19-s + 4.50·21-s + i·23-s + (3.75 + 3.29i)25-s − 5.53i·27-s + 6.61·29-s − 8.29·31-s + ⋯ |
L(s) = 1 | − 0.914i·3-s + (−0.935 − 0.352i)5-s + 1.07i·7-s + 0.164·9-s + 0.598·11-s + 1.30i·13-s + (−0.321 + 0.855i)15-s − 0.768i·17-s + 1.41·19-s + 0.983·21-s + 0.208i·23-s + (0.751 + 0.659i)25-s − 1.06i·27-s + 1.22·29-s − 1.49·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 + 0.352i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.935 + 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.42666 - 0.259573i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.42666 - 0.259573i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (2.09 + 0.787i)T \) |
| 23 | \( 1 - iT \) |
good | 3 | \( 1 + 1.58iT - 3T^{2} \) |
| 7 | \( 1 - 2.84iT - 7T^{2} \) |
| 11 | \( 1 - 1.98T + 11T^{2} \) |
| 13 | \( 1 - 4.69iT - 13T^{2} \) |
| 17 | \( 1 + 3.16iT - 17T^{2} \) |
| 19 | \( 1 - 6.16T + 19T^{2} \) |
| 29 | \( 1 - 6.61T + 29T^{2} \) |
| 31 | \( 1 + 8.29T + 31T^{2} \) |
| 37 | \( 1 - 1.71iT - 37T^{2} \) |
| 41 | \( 1 - 6.72T + 41T^{2} \) |
| 43 | \( 1 + 0.177iT - 43T^{2} \) |
| 47 | \( 1 + 11.4iT - 47T^{2} \) |
| 53 | \( 1 - 6.18iT - 53T^{2} \) |
| 59 | \( 1 - 7.61T + 59T^{2} \) |
| 61 | \( 1 - 11.8T + 61T^{2} \) |
| 67 | \( 1 + 3.07iT - 67T^{2} \) |
| 71 | \( 1 + 1.96T + 71T^{2} \) |
| 73 | \( 1 - 4.94iT - 73T^{2} \) |
| 79 | \( 1 + 8.53T + 79T^{2} \) |
| 83 | \( 1 - 3.49iT - 83T^{2} \) |
| 89 | \( 1 + 7.07T + 89T^{2} \) |
| 97 | \( 1 - 7.00iT - 97T^{2} \) |
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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
−9.779970358277527884084430673241, −9.052395804338948081072777799072, −8.386620613051651416734956582018, −7.26607220586101610342826468654, −6.96183588175510068362863998971, −5.74214391672062257282862318281, −4.73455067530039023568847401227, −3.66898631213386309527397595504, −2.32738246876213597503285161610, −1.08820745558345285701988489394,
0.942123567764900871550747742166, 3.15621198229676692687130117766, 3.83256189385994783000852600614, 4.53575226444549353048139604048, 5.64789785064034266795479185859, 6.96204873753334986858541602404, 7.54853343654295636218222502624, 8.396201465818514232338389856609, 9.505757503565795834296850996530, 10.27069144141292646923100880800