L(s) = 1 | − 1.40·2-s − 2.91·3-s − 2.02·4-s + 4.10·6-s − 11.7i·7-s + 8.46·8-s − 0.493·9-s − 6.36·11-s + 5.89·12-s + 17.7·13-s + 16.4i·14-s − 3.82·16-s − 17.5i·17-s + 0.694·18-s + (16.8 + 8.86i)19-s + ⋯ |
L(s) = 1 | − 0.703·2-s − 0.972·3-s − 0.505·4-s + 0.683·6-s − 1.67i·7-s + 1.05·8-s − 0.0548·9-s − 0.578·11-s + 0.491·12-s + 1.36·13-s + 1.17i·14-s − 0.238·16-s − 1.03i·17-s + 0.0385·18-s + (0.884 + 0.466i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0216i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 - 0.0216i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2961105078\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2961105078\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 + (-16.8 - 8.86i)T \) |
good | 2 | \( 1 + 1.40T + 4T^{2} \) |
| 3 | \( 1 + 2.91T + 9T^{2} \) |
| 7 | \( 1 + 11.7iT - 49T^{2} \) |
| 11 | \( 1 + 6.36T + 121T^{2} \) |
| 13 | \( 1 - 17.7T + 169T^{2} \) |
| 17 | \( 1 + 17.5iT - 289T^{2} \) |
| 23 | \( 1 + 1.53iT - 529T^{2} \) |
| 29 | \( 1 + 39.9iT - 841T^{2} \) |
| 31 | \( 1 + 9.24iT - 961T^{2} \) |
| 37 | \( 1 + 56.4T + 1.36e3T^{2} \) |
| 41 | \( 1 - 39.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 18.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 61.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 66.5T + 2.80e3T^{2} \) |
| 59 | \( 1 + 9.42iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 36.5T + 3.72e3T^{2} \) |
| 67 | \( 1 + 2.55T + 4.48e3T^{2} \) |
| 71 | \( 1 - 53.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 115. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 95.9iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 143. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 19.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 163.T + 9.40e3T^{2} \) |
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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
−10.35459384393983568624564758408, −9.708817955519510548107953563160, −8.473785834584043461700576720473, −7.67574461364251294933161119459, −6.78369894668458822965059711992, −5.57414229542231804185364881985, −4.61341445443750405707649436692, −3.55608127811109941824922428734, −1.13592898766220798956471221151, −0.22445372911523342126252431450,
1.49787555494969014744063165759, 3.26196092729988397595541805245, 4.95793975334208932961519234647, 5.57477860561879399549145459797, 6.43572400170674150623211302410, 7.918543890046391202702081265268, 8.768087577193326029383177901572, 9.186940189969917591764684244201, 10.57290311153427847583981140191, 10.97166511738185151294981985151