L(s) = 1 | + (0.366 + 0.633i)2-s + (0.732 − 1.26i)4-s + (0.5 − 0.866i)5-s + (2.36 + 4.09i)7-s + 2.53·8-s + 0.732·10-s + (−2.86 − 4.96i)11-s + (−0.732 + 1.26i)13-s + (−1.73 + 3i)14-s + (−0.535 − 0.928i)16-s + 2.73·17-s + 4.46·19-s + (−0.732 − 1.26i)20-s + (2.09 − 3.63i)22-s + (−1.73 + 3i)23-s + ⋯ |
L(s) = 1 | + (0.258 + 0.448i)2-s + (0.366 − 0.633i)4-s + (0.223 − 0.387i)5-s + (0.894 + 1.54i)7-s + 0.896·8-s + 0.231·10-s + (−0.864 − 1.49i)11-s + (−0.203 + 0.351i)13-s + (−0.462 + 0.801i)14-s + (−0.133 − 0.232i)16-s + 0.662·17-s + 1.02·19-s + (−0.163 − 0.283i)20-s + (0.447 − 0.774i)22-s + (−0.361 + 0.625i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.92412 + 0.168338i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.92412 + 0.168338i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
good | 2 | \( 1 + (-0.366 - 0.633i)T + (-1 + 1.73i)T^{2} \) |
| 7 | \( 1 + (-2.36 - 4.09i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.86 + 4.96i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.732 - 1.26i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 2.73T + 17T^{2} \) |
| 19 | \( 1 - 4.46T + 19T^{2} \) |
| 23 | \( 1 + (1.73 - 3i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.59 - 2.76i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.5 + 2.59i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 2.73T + 37T^{2} \) |
| 41 | \( 1 + (3.59 - 6.23i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (0.0980 + 0.169i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (4.36 + 7.56i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 6.73T + 53T^{2} \) |
| 59 | \( 1 + (4.13 - 7.16i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2 + 3.46i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.73 - 3i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 3.73T + 71T^{2} \) |
| 73 | \( 1 + 7.66T + 73T^{2} \) |
| 79 | \( 1 + (7.73 + 13.3i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.09 - 1.90i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 5.19T + 89T^{2} \) |
| 97 | \( 1 + (-4.83 - 8.36i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.49101144858109172841332660552, −10.41241874163121413025356255186, −9.377882143004051183326027534374, −8.411473963649291361827553767377, −7.67892939728253523283886597651, −6.17244297190266264245776517807, −5.47812845299257170782773362205, −4.98604381478938733878371296454, −2.93759745194110355502911529878, −1.56852296402028340279410962845,
1.67580234953427853447892219728, 3.01520316115057184064501069483, 4.24866540564882009891139629645, 5.05517165695887704667901090559, 6.86752763437159531934474806987, 7.55711850360958116640063782291, 8.035331299489770705053724023178, 9.928308657902474119430882027611, 10.37791830239912004971699963908, 11.16624559026033408609084226763