L(s) = 1 | + 2.17·2-s − 1.39i·3-s + 2.73·4-s + 5-s − 3.03i·6-s − 0.972i·7-s + 1.58·8-s + 1.05·9-s + 2.17·10-s + (−1.20 − 3.08i)11-s − 3.81i·12-s + 5.65·13-s − 2.11i·14-s − 1.39i·15-s − 2.00·16-s + 4.67i·17-s + ⋯ |
L(s) = 1 | + 1.53·2-s − 0.806i·3-s + 1.36·4-s + 0.447·5-s − 1.23i·6-s − 0.367i·7-s + 0.561·8-s + 0.350·9-s + 0.687·10-s + (−0.364 − 0.931i)11-s − 1.10i·12-s + 1.56·13-s − 0.565i·14-s − 0.360i·15-s − 0.501·16-s + 1.13i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.495 + 0.868i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.495 + 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.093698380\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.093698380\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 11 | \( 1 + (1.20 + 3.08i)T \) |
| 19 | \( 1 + (4.31 - 0.631i)T \) |
good | 2 | \( 1 - 2.17T + 2T^{2} \) |
| 3 | \( 1 + 1.39iT - 3T^{2} \) |
| 7 | \( 1 + 0.972iT - 7T^{2} \) |
| 13 | \( 1 - 5.65T + 13T^{2} \) |
| 17 | \( 1 - 4.67iT - 17T^{2} \) |
| 23 | \( 1 + 1.62T + 23T^{2} \) |
| 29 | \( 1 - 6.29T + 29T^{2} \) |
| 31 | \( 1 + 8.09iT - 31T^{2} \) |
| 37 | \( 1 - 4.08iT - 37T^{2} \) |
| 41 | \( 1 + 10.2T + 41T^{2} \) |
| 43 | \( 1 - 9.06iT - 43T^{2} \) |
| 47 | \( 1 - 4.98T + 47T^{2} \) |
| 53 | \( 1 + 1.30iT - 53T^{2} \) |
| 59 | \( 1 + 3.03iT - 59T^{2} \) |
| 61 | \( 1 - 12.2iT - 61T^{2} \) |
| 67 | \( 1 + 8.46iT - 67T^{2} \) |
| 71 | \( 1 - 15.6iT - 71T^{2} \) |
| 73 | \( 1 - 3.75iT - 73T^{2} \) |
| 79 | \( 1 + 0.484T + 79T^{2} \) |
| 83 | \( 1 - 7.80iT - 83T^{2} \) |
| 89 | \( 1 - 7.55iT - 89T^{2} \) |
| 97 | \( 1 + 9.02iT - 97T^{2} \) |
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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
−10.10232511611944751721749280329, −8.620377086317705784268867844929, −8.072550689945784561666272379082, −6.73785549871221421300082834936, −6.23558285015666218038813780490, −5.69505814415722859708084148345, −4.34929020690396570879525192052, −3.70921016700166389926507738437, −2.50574130745493624114683527683, −1.30098203523612359775075221957,
1.94839299949362856759460319667, 3.11483665356098645628131835527, 4.03418663789745865161397542958, 4.81313796504149106533074499803, 5.42106473023035418557817810824, 6.43930414372522704482681135640, 7.09640850296682300685798394279, 8.607515338382678832581646486365, 9.258850614823155622770340083699, 10.39174238828628643851758041315