| L(s) = 1 | + (−1.67 − 0.117i)2-s + (−0.0321 − 0.921i)3-s + (0.808 + 0.113i)4-s + (1.70 − 1.45i)5-s + (−0.0539 + 1.54i)6-s + (−0.0328 − 0.0568i)7-s + (1.94 + 0.412i)8-s + (2.14 − 0.150i)9-s + (−3.01 + 2.23i)10-s + (0.442 + 4.20i)11-s + (0.0786 − 0.748i)12-s + (1.67 + 2.48i)13-s + (0.0482 + 0.0989i)14-s + (−1.39 − 1.52i)15-s + (−4.77 − 1.36i)16-s + (0.694 − 1.71i)17-s + ⋯ |
| L(s) = 1 | + (−1.18 − 0.0827i)2-s + (−0.0185 − 0.531i)3-s + (0.404 + 0.0568i)4-s + (0.760 − 0.648i)5-s + (−0.0220 + 0.631i)6-s + (−0.0124 − 0.0214i)7-s + (0.686 + 0.145i)8-s + (0.715 − 0.0500i)9-s + (−0.954 + 0.705i)10-s + (0.133 + 1.26i)11-s + (0.0227 − 0.216i)12-s + (0.465 + 0.690i)13-s + (0.0129 + 0.0264i)14-s + (−0.359 − 0.392i)15-s + (−1.19 − 0.342i)16-s + (0.168 − 0.416i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.842 + 0.538i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.842 + 0.538i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.893925 - 0.261181i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.893925 - 0.261181i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.70 + 1.45i)T \) |
| 19 | \( 1 + (-2.33 - 3.67i)T \) |
| good | 2 | \( 1 + (1.67 + 0.117i)T + (1.98 + 0.278i)T^{2} \) |
| 3 | \( 1 + (0.0321 + 0.921i)T + (-2.99 + 0.209i)T^{2} \) |
| 7 | \( 1 + (0.0328 + 0.0568i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.442 - 4.20i)T + (-10.7 + 2.28i)T^{2} \) |
| 13 | \( 1 + (-1.67 - 2.48i)T + (-4.86 + 12.0i)T^{2} \) |
| 17 | \( 1 + (-0.694 + 1.71i)T + (-12.2 - 11.8i)T^{2} \) |
| 23 | \( 1 + (0.519 + 0.501i)T + (0.802 + 22.9i)T^{2} \) |
| 29 | \( 1 + (-3.49 - 8.65i)T + (-20.8 + 20.1i)T^{2} \) |
| 31 | \( 1 + (-0.593 + 0.659i)T + (-3.24 - 30.8i)T^{2} \) |
| 37 | \( 1 + (-3.24 - 2.36i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (4.39 + 1.26i)T + (34.7 + 21.7i)T^{2} \) |
| 43 | \( 1 + (-1.80 + 10.2i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (1.82 + 4.50i)T + (-33.8 + 32.6i)T^{2} \) |
| 53 | \( 1 + (-3.54 - 0.498i)T + (50.9 + 14.6i)T^{2} \) |
| 59 | \( 1 + (-0.952 - 3.81i)T + (-52.0 + 27.6i)T^{2} \) |
| 61 | \( 1 + (-4.90 - 4.73i)T + (2.12 + 60.9i)T^{2} \) |
| 67 | \( 1 + (6.03 + 3.77i)T + (29.3 + 60.2i)T^{2} \) |
| 71 | \( 1 + (10.7 + 5.72i)T + (39.7 + 58.8i)T^{2} \) |
| 73 | \( 1 + (-8.25 + 12.2i)T + (-27.3 - 67.6i)T^{2} \) |
| 79 | \( 1 + (-0.0469 - 1.34i)T + (-78.8 + 5.51i)T^{2} \) |
| 83 | \( 1 + (9.35 - 10.3i)T + (-8.67 - 82.5i)T^{2} \) |
| 89 | \( 1 + (2.63 - 0.754i)T + (75.4 - 47.1i)T^{2} \) |
| 97 | \( 1 + (10.1 - 6.33i)T + (42.5 - 87.1i)T^{2} \) |
| 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
−10.42288364977315334212424897908, −9.954539622810451954333628001921, −9.182809505727016975335516976007, −8.421350010409776813527528560195, −7.32972155125827847421504463802, −6.72402588236997617338399109901, −5.24726275005653460320907986663, −4.23490604498382503073183022002, −1.99222739171753537890745210680, −1.23928310058644737617939920648,
1.15232014382085710470763370322, 2.90415571911014893053632133463, 4.24433552191805486480665205125, 5.63728919158290820492594739265, 6.59385852603386520774361208454, 7.68722662551510832986216445805, 8.526265575249822602940728687186, 9.508282524315188461855257957958, 9.995151089023533086748517768784, 10.80298273049070294076386351906
