L(s) = 1 | + (0.222 + 0.385i)3-s + (−0.5 + 0.866i)4-s + 1.80·5-s + (0.400 − 0.694i)9-s + (0.5 + 0.866i)11-s − 0.445·12-s + (0.400 + 0.694i)15-s + (−0.499 − 0.866i)16-s + (−0.900 + 1.56i)20-s + (−0.623 − 1.07i)23-s + 2.24·25-s + 0.801·27-s − 1.24·31-s + (−0.222 + 0.385i)33-s + (0.400 + 0.694i)36-s + (−0.222 − 0.385i)37-s + ⋯ |
L(s) = 1 | + (0.222 + 0.385i)3-s + (−0.5 + 0.866i)4-s + 1.80·5-s + (0.400 − 0.694i)9-s + (0.5 + 0.866i)11-s − 0.445·12-s + (0.400 + 0.694i)15-s + (−0.499 − 0.866i)16-s + (−0.900 + 1.56i)20-s + (−0.623 − 1.07i)23-s + 2.24·25-s + 0.801·27-s − 1.24·31-s + (−0.222 + 0.385i)33-s + (0.400 + 0.694i)36-s + (−0.222 − 0.385i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.668 - 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.668 - 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.555091241\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.555091241\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 3 | \( 1 + (-0.222 - 0.385i)T + (-0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 - 1.80T + T^{2} \) |
| 7 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.623 + 1.07i)T + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + 1.24T + T^{2} \) |
| 37 | \( 1 + (0.222 + 0.385i)T + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + 1.24T + T^{2} \) |
| 53 | \( 1 + 1.80T + T^{2} \) |
| 59 | \( 1 + (0.222 - 0.385i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (0.222 - 0.385i)T + (-0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (-0.623 - 1.07i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (0.900 - 1.56i)T + (-0.5 - 0.866i)T^{2} \) |
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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
−9.557533532412402410412094291256, −8.999513583682295134449595311057, −8.182376216818277910501064590618, −6.95979408917485322469852805493, −6.50643633052932050019644965245, −5.41081938756247366711609829591, −4.57585311094722520186872946008, −3.74213256637403067702302565758, −2.66320411705426773043759822256, −1.65550912931061793235162238534,
1.47573279448361349308033880185, 1.90621716319244380771349774574, 3.27803989427556190133315264290, 4.70851902956968859427785684825, 5.40685512322721334723051482898, 6.09079546454994610816447797243, 6.65514215230311615052708103761, 7.83822822853876985496018895545, 8.755102995600120555998996211888, 9.505665489637455148315344149447