L(s) = 1 | − 3-s + 1.73i·5-s + 9-s + 5.19i·11-s + 6.92i·13-s − 1.73i·15-s + 3.46i·17-s + 2·19-s − 6.92i·23-s + 2.00·25-s − 27-s − 9·29-s + 31-s − 5.19i·33-s − 2·37-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.774i·5-s + 0.333·9-s + 1.56i·11-s + 1.92i·13-s − 0.447i·15-s + 0.840i·17-s + 0.458·19-s − 1.44i·23-s + 0.400·25-s − 0.192·27-s − 1.67·29-s + 0.179·31-s − 0.904i·33-s − 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 - 0.327i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.944 - 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9797051281\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9797051281\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 1.73iT - 5T^{2} \) |
| 11 | \( 1 - 5.19iT - 11T^{2} \) |
| 13 | \( 1 - 6.92iT - 13T^{2} \) |
| 17 | \( 1 - 3.46iT - 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + 6.92iT - 23T^{2} \) |
| 29 | \( 1 + 9T + 29T^{2} \) |
| 31 | \( 1 - T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 3.46iT - 41T^{2} \) |
| 43 | \( 1 + 3.46iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 9T + 53T^{2} \) |
| 59 | \( 1 + 3T + 59T^{2} \) |
| 61 | \( 1 - 6.92iT - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 6.92iT - 71T^{2} \) |
| 73 | \( 1 - 6.92iT - 73T^{2} \) |
| 79 | \( 1 - 1.73iT - 79T^{2} \) |
| 83 | \( 1 + 15T + 83T^{2} \) |
| 89 | \( 1 - 10.3iT - 89T^{2} \) |
| 97 | \( 1 - 8.66iT - 97T^{2} \) |
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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
−9.399313986225159804161803806916, −8.689566754130849929814897972891, −7.44570103384186901012538675461, −6.94111526070505957458140205928, −6.46142402800890392576405098808, −5.41254595532140251450104078766, −4.41899804655451168354070894288, −3.91557220328210265752888404067, −2.41829803001243573852728731649, −1.66366257881429197187179758114,
0.38878806783810151063413324181, 1.24236527673125871696258896508, 2.96422302701552219325839491158, 3.62439317450712667913210896472, 4.95379099453019211167493133182, 5.54110775059362398383594301348, 5.93059242114323604959642214856, 7.23480578719337984755746912612, 7.87714954488260912208687688887, 8.619054464138582699759206127593