L(s) = 1 | + (0.0893 + 0.0893i)5-s − 7-s + (2.42 − 2.42i)11-s + (−3.86 − 3.86i)13-s − 0.794i·17-s + (2.65 − 2.65i)19-s − 3.92i·23-s − 4.98i·25-s + (−7.47 + 7.47i)29-s + 5.55i·31-s + (−0.0893 − 0.0893i)35-s + (−6.35 + 6.35i)37-s − 6.96·41-s + (1.25 + 1.25i)43-s + 6.48·47-s + ⋯ |
L(s) = 1 | + (0.0399 + 0.0399i)5-s − 0.377·7-s + (0.731 − 0.731i)11-s + (−1.07 − 1.07i)13-s − 0.192i·17-s + (0.608 − 0.608i)19-s − 0.817i·23-s − 0.996i·25-s + (−1.38 + 1.38i)29-s + 0.997i·31-s + (−0.0151 − 0.0151i)35-s + (−1.04 + 1.04i)37-s − 1.08·41-s + (0.191 + 0.191i)43-s + 0.945·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00496i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.00496i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3587829950\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3587829950\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 + (-0.0893 - 0.0893i)T + 5iT^{2} \) |
| 11 | \( 1 + (-2.42 + 2.42i)T - 11iT^{2} \) |
| 13 | \( 1 + (3.86 + 3.86i)T + 13iT^{2} \) |
| 17 | \( 1 + 0.794iT - 17T^{2} \) |
| 19 | \( 1 + (-2.65 + 2.65i)T - 19iT^{2} \) |
| 23 | \( 1 + 3.92iT - 23T^{2} \) |
| 29 | \( 1 + (7.47 - 7.47i)T - 29iT^{2} \) |
| 31 | \( 1 - 5.55iT - 31T^{2} \) |
| 37 | \( 1 + (6.35 - 6.35i)T - 37iT^{2} \) |
| 41 | \( 1 + 6.96T + 41T^{2} \) |
| 43 | \( 1 + (-1.25 - 1.25i)T + 43iT^{2} \) |
| 47 | \( 1 - 6.48T + 47T^{2} \) |
| 53 | \( 1 + (-0.620 - 0.620i)T + 53iT^{2} \) |
| 59 | \( 1 + (3.39 - 3.39i)T - 59iT^{2} \) |
| 61 | \( 1 + (-7.51 - 7.51i)T + 61iT^{2} \) |
| 67 | \( 1 + (2.22 - 2.22i)T - 67iT^{2} \) |
| 71 | \( 1 + 7.95iT - 71T^{2} \) |
| 73 | \( 1 - 12.9iT - 73T^{2} \) |
| 79 | \( 1 - 10.1iT - 79T^{2} \) |
| 83 | \( 1 + (11.2 + 11.2i)T + 83iT^{2} \) |
| 89 | \( 1 + 5.52T + 89T^{2} \) |
| 97 | \( 1 + 4.33T + 97T^{2} \) |
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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
−8.228410154510407681720002099476, −7.00211720934420719493941539040, −6.94541334594460040103312927981, −5.70945973231814285174722695312, −5.22414299407557365048236030421, −4.25753846902073955341714670152, −3.21832689375428929590443473424, −2.72236725968002332433151125407, −1.30879871505852515714576566788, −0.10108859039651993587865423513,
1.61056982477162599350394124975, 2.27851801938706190456884049902, 3.64918983280735603209408246564, 4.07784826325104794929385981288, 5.15305269686634271993062660658, 5.79920110579522293786660339291, 6.76574856340606910587593863154, 7.30703774716235706119488042949, 7.88563265340242269604506904505, 9.108087121641443717853145341253