L(s) = 1 | − 1.52i·2-s − 2.99·3-s − 0.320·4-s − 2.94i·5-s + 4.56i·6-s − 2.55i·8-s + 5.97·9-s − 4.48·10-s − 2.37i·11-s + 0.960·12-s + (−2.32 − 2.75i)13-s + 8.82i·15-s − 4.53·16-s + 5.36·17-s − 9.09i·18-s − 5.35i·19-s + ⋯ |
L(s) = 1 | − 1.07i·2-s − 1.72·3-s − 0.160·4-s − 1.31i·5-s + 1.86i·6-s − 0.904i·8-s + 1.99·9-s − 1.41·10-s − 0.714i·11-s + 0.277·12-s + (−0.645 − 0.763i)13-s + 2.27i·15-s − 1.13·16-s + 1.30·17-s − 2.14i·18-s − 1.22i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.645 - 0.763i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.645 - 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.289457 + 0.623616i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.289457 + 0.623616i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (2.32 + 2.75i)T \) |
good | 2 | \( 1 + 1.52iT - 2T^{2} \) |
| 3 | \( 1 + 2.99T + 3T^{2} \) |
| 5 | \( 1 + 2.94iT - 5T^{2} \) |
| 11 | \( 1 + 2.37iT - 11T^{2} \) |
| 17 | \( 1 - 5.36T + 17T^{2} \) |
| 19 | \( 1 + 5.35iT - 19T^{2} \) |
| 23 | \( 1 + 2.79T + 23T^{2} \) |
| 29 | \( 1 - 0.585T + 29T^{2} \) |
| 31 | \( 1 - 8.33iT - 31T^{2} \) |
| 37 | \( 1 - 0.675iT - 37T^{2} \) |
| 41 | \( 1 - 11.2iT - 41T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 + 0.537iT - 47T^{2} \) |
| 53 | \( 1 - 7.90T + 53T^{2} \) |
| 59 | \( 1 - 6.14iT - 59T^{2} \) |
| 61 | \( 1 - 2.78T + 61T^{2} \) |
| 67 | \( 1 + 4.21iT - 67T^{2} \) |
| 71 | \( 1 + 1.25iT - 71T^{2} \) |
| 73 | \( 1 + 10.8iT - 73T^{2} \) |
| 79 | \( 1 - 6.29T + 79T^{2} \) |
| 83 | \( 1 + 1.74iT - 83T^{2} \) |
| 89 | \( 1 + 10.2iT - 89T^{2} \) |
| 97 | \( 1 + 9.90iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.26504987363595948434932732859, −9.651017586473316370098382780006, −8.442116437825226204933326090257, −7.20540771539575097797837512044, −6.15843863274573704549325127597, −5.21251613197756254811406822282, −4.64900083251488963828144755298, −3.20607863607314476209728323246, −1.34021620224077051654192141902, −0.50104839014721444935930035914,
2.08765232840333561431705794195, 3.97474364887153135891769869466, 5.25751558608009591684026059713, 5.87126493416648098712859739982, 6.70491492981965769401793624077, 7.17480399559716060604164840871, 7.958893889917902209459179247349, 9.797182224413888887488803996786, 10.28152732268734751426847361070, 11.17667430137357220173116756214