L(s) = 1 | + (0.5 + 0.866i)3-s + 4-s + (−0.499 + 0.866i)9-s + (0.5 + 0.866i)12-s + (0.5 − 0.866i)13-s + 16-s + (−0.5 + 0.866i)19-s + (−0.5 + 0.866i)25-s − 0.999·27-s + (1 − 1.73i)31-s + (−0.499 + 0.866i)36-s − 37-s + 0.999·39-s + (0.5 + 0.866i)43-s + (0.5 + 0.866i)48-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)3-s + 4-s + (−0.499 + 0.866i)9-s + (0.5 + 0.866i)12-s + (0.5 − 0.866i)13-s + 16-s + (−0.5 + 0.866i)19-s + (−0.5 + 0.866i)25-s − 0.999·27-s + (1 − 1.73i)31-s + (−0.499 + 0.866i)36-s − 37-s + 0.999·39-s + (0.5 + 0.866i)43-s + (0.5 + 0.866i)48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.617 - 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.617 - 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.696296276\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.696296276\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + (-0.5 + 0.866i)T \) |
good | 2 | \( 1 - T^{2} \) |
| 5 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
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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
−9.683658150700785357083403861823, −8.658113727467504118705857677477, −7.954417759635348774516630012654, −7.38917850467272199772065136890, −6.08638072854603139718020848497, −5.70125092543156735846796610153, −4.47847902078195907124749428176, −3.54268037307200548863143341021, −2.81365633739104343889341931335, −1.73041671130129816161272089754,
1.35722756020361476129655897864, 2.28814900804269239706097991249, 3.10631121553068922496668328592, 4.18042676357661467957507815636, 5.49893730684909701658459104973, 6.57904853194271840348577926976, 6.71801570451689836103093637457, 7.63236998634976486279449406144, 8.492599284537430327152719892486, 8.996826493859654810548368651477