L(s) = 1 | + (−0.592 − 0.342i)7-s + (0.326 − 0.118i)13-s + (0.5 − 0.866i)19-s + (0.939 − 0.342i)25-s + (−1.11 − 0.642i)31-s + 1.53·37-s + (1.26 − 0.223i)43-s + (−0.266 − 0.460i)49-s + (0.0603 − 0.342i)61-s + (0.439 − 0.524i)67-s + (0.326 + 0.118i)73-s + (0.233 − 0.642i)79-s + (−0.233 − 0.0412i)91-s + (0.766 − 0.642i)97-s + (−1.70 + 0.984i)103-s + ⋯ |
L(s) = 1 | + (−0.592 − 0.342i)7-s + (0.326 − 0.118i)13-s + (0.5 − 0.866i)19-s + (0.939 − 0.342i)25-s + (−1.11 − 0.642i)31-s + 1.53·37-s + (1.26 − 0.223i)43-s + (−0.266 − 0.460i)49-s + (0.0603 − 0.342i)61-s + (0.439 − 0.524i)67-s + (0.326 + 0.118i)73-s + (0.233 − 0.642i)79-s + (−0.233 − 0.0412i)91-s + (0.766 − 0.642i)97-s + (−1.70 + 0.984i)103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.756 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.756 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.137507613\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.137507613\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-0.5 + 0.866i)T \) |
good | 5 | \( 1 + (-0.939 + 0.342i)T^{2} \) |
| 7 | \( 1 + (0.592 + 0.342i)T + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.326 + 0.118i)T + (0.766 - 0.642i)T^{2} \) |
| 17 | \( 1 + (0.173 - 0.984i)T^{2} \) |
| 23 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 29 | \( 1 + (0.173 + 0.984i)T^{2} \) |
| 31 | \( 1 + (1.11 + 0.642i)T + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - 1.53T + T^{2} \) |
| 41 | \( 1 + (0.766 + 0.642i)T^{2} \) |
| 43 | \( 1 + (-1.26 + 0.223i)T + (0.939 - 0.342i)T^{2} \) |
| 47 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 53 | \( 1 + (-0.939 - 0.342i)T^{2} \) |
| 59 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 61 | \( 1 + (-0.0603 + 0.342i)T + (-0.939 - 0.342i)T^{2} \) |
| 67 | \( 1 + (-0.439 + 0.524i)T + (-0.173 - 0.984i)T^{2} \) |
| 71 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 73 | \( 1 + (-0.326 - 0.118i)T + (0.766 + 0.642i)T^{2} \) |
| 79 | \( 1 + (-0.233 + 0.642i)T + (-0.766 - 0.642i)T^{2} \) |
| 83 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (0.766 - 0.642i)T^{2} \) |
| 97 | \( 1 + (-0.766 + 0.642i)T + (0.173 - 0.984i)T^{2} \) |
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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
−9.107420199735405275671523380291, −8.114230768003278819796456581509, −7.37724814627203980488948904250, −6.66486898434618355981375049220, −5.92130306300428571782473227279, −5.02085762841085533229893890530, −4.11859874466801241384100931819, −3.25943900846408191559638608243, −2.33910994776101595092448930334, −0.836938239494087750550743392080,
1.27922242948173398644991721308, 2.57761491931209092971974309877, 3.42187450396559928580983996589, 4.29419107931578137219567759459, 5.36494422859218111114762113446, 5.99078759114862029209654087002, 6.79759073898010487548091335938, 7.58144956104003611126479901307, 8.362430086959947009880632574405, 9.222837292315297051375742273993