L(s) = 1 | − 1.78i·2-s + 2.38i·3-s − 1.18·4-s + (−1.16 − 1.90i)5-s + 4.25·6-s + 4.23i·7-s − 1.45i·8-s − 2.68·9-s + (−3.40 + 2.07i)10-s − 0.490·11-s − 2.82i·12-s + 4.16i·13-s + 7.56·14-s + (4.54 − 2.77i)15-s − 4.96·16-s − 2.03i·17-s + ⋯ |
L(s) = 1 | − 1.26i·2-s + 1.37i·3-s − 0.592·4-s + (−0.521 − 0.853i)5-s + 1.73·6-s + 1.60i·7-s − 0.514i·8-s − 0.894·9-s + (−1.07 + 0.657i)10-s − 0.148·11-s − 0.815i·12-s + 1.15i·13-s + 2.02·14-s + (1.17 − 0.717i)15-s − 1.24·16-s − 0.493i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.521 - 0.853i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.521 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5236201383\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5236201383\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (1.16 + 1.90i)T \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + 1.78iT - 2T^{2} \) |
| 3 | \( 1 - 2.38iT - 3T^{2} \) |
| 7 | \( 1 - 4.23iT - 7T^{2} \) |
| 11 | \( 1 + 0.490T + 11T^{2} \) |
| 13 | \( 1 - 4.16iT - 13T^{2} \) |
| 17 | \( 1 + 2.03iT - 17T^{2} \) |
| 23 | \( 1 + 4.39iT - 23T^{2} \) |
| 29 | \( 1 + 3.26T + 29T^{2} \) |
| 31 | \( 1 + 4.08T + 31T^{2} \) |
| 37 | \( 1 - 2.14iT - 37T^{2} \) |
| 41 | \( 1 + 4.36T + 41T^{2} \) |
| 43 | \( 1 + 10.6iT - 43T^{2} \) |
| 47 | \( 1 - 2.62iT - 47T^{2} \) |
| 53 | \( 1 - 11.4iT - 53T^{2} \) |
| 59 | \( 1 + 0.542T + 59T^{2} \) |
| 61 | \( 1 + 13.6T + 61T^{2} \) |
| 67 | \( 1 - 7.15iT - 67T^{2} \) |
| 71 | \( 1 + 6.03T + 71T^{2} \) |
| 73 | \( 1 + 2.05iT - 73T^{2} \) |
| 79 | \( 1 - 5.34T + 79T^{2} \) |
| 83 | \( 1 + 8.11iT - 83T^{2} \) |
| 89 | \( 1 + 4.34T + 89T^{2} \) |
| 97 | \( 1 + 5.64iT - 97T^{2} \) |
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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
−9.432543904607985097818807819853, −9.067410971960165252086867157210, −8.648229895838816204128851142935, −7.26772844263139775828262856804, −5.99261640724415389635612094230, −5.04250924372314609863244187104, −4.45268695483722842906252127263, −3.66429345390501108463003381057, −2.72075416182507882695818348704, −1.71987409738475356594434022616,
0.18484494887276242103631191401, 1.64993745628663602122471163360, 3.04126294058688980007734257962, 4.02821886222184393034297636264, 5.29997391897351687730431032919, 6.25168228655433338495318912148, 6.78825237481442510625593381251, 7.51399067921964481473637007804, 7.70662075358171577491861983986, 8.303573537702020033893010453783