L(s) = 1 | − 3.60·2-s − 3.60·3-s + 8.99·4-s + 12.9·6-s + 5i·7-s − 18.0·8-s + 3.99·9-s − 10·11-s − 32.4·12-s + 3.60·13-s − 18.0i·14-s + 28.9·16-s − 15i·17-s − 14.4·18-s + (6 + 18.0i)19-s + ⋯ |
L(s) = 1 | − 1.80·2-s − 1.20·3-s + 2.24·4-s + 2.16·6-s + 0.714i·7-s − 2.25·8-s + 0.444·9-s − 0.909·11-s − 2.70·12-s + 0.277·13-s − 1.28i·14-s + 1.81·16-s − 0.882i·17-s − 0.801·18-s + (0.315 + 0.948i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.706i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 + 0.706i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.01631908791\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.01631908791\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 + (-6 - 18.0i)T \) |
good | 2 | \( 1 + 3.60T + 4T^{2} \) |
| 3 | \( 1 + 3.60T + 9T^{2} \) |
| 7 | \( 1 - 5iT - 49T^{2} \) |
| 11 | \( 1 + 10T + 121T^{2} \) |
| 13 | \( 1 - 3.60T + 169T^{2} \) |
| 17 | \( 1 + 15iT - 289T^{2} \) |
| 23 | \( 1 - 35iT - 529T^{2} \) |
| 29 | \( 1 - 18.0iT - 841T^{2} \) |
| 31 | \( 1 - 36.0iT - 961T^{2} \) |
| 37 | \( 1 - 21.6T + 1.36e3T^{2} \) |
| 41 | \( 1 + 36.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 20iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 10iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 75.7T + 2.80e3T^{2} \) |
| 59 | \( 1 - 18.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 40T + 3.72e3T^{2} \) |
| 67 | \( 1 + 39.6T + 4.48e3T^{2} \) |
| 71 | \( 1 + 108. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 105iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 36.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 40iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 + 122.T + 9.40e3T^{2} \) |
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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
−10.47896553894778576225646202938, −9.598890007182833990635558003775, −8.780630697482239504678541027590, −7.81485808785815738985063579877, −7.00410662989596097938371708543, −5.90431373673978617743100352714, −5.23139667101396254989610206503, −2.96031240123102915409498675116, −1.47860304651263597695840423328, −0.01869733813459852836241354637,
0.952712733363147178738686937538, 2.60000399363871039369402658868, 4.57106977885435652762082776236, 5.99123162871928463389035959198, 6.64251629629239723213848085928, 7.68843015395709665593182984573, 8.365874871856535059670836540997, 9.517314050706988636216781907405, 10.36371845707663486946939623915, 10.90426454694618691108877229564