L(s) = 1 | + (0.766 + 0.642i)2-s + (3.03 − 1.10i)3-s + (0.173 + 0.984i)4-s + (3.03 + 1.10i)6-s + (1.36 + 2.35i)7-s + (−0.500 + 0.866i)8-s + (5.67 − 4.75i)9-s + (−1.18 + 2.05i)11-s + (1.61 + 2.79i)12-s + (0.0945 + 0.0344i)13-s + (−0.472 + 2.68i)14-s + (−0.939 + 0.342i)16-s + (−4.17 − 3.50i)17-s + 7.40·18-s + (1.18 + 4.19i)19-s + ⋯ |
L(s) = 1 | + (0.541 + 0.454i)2-s + (1.74 − 0.636i)3-s + (0.0868 + 0.492i)4-s + (1.23 + 0.450i)6-s + (0.514 + 0.891i)7-s + (−0.176 + 0.306i)8-s + (1.89 − 1.58i)9-s + (−0.358 + 0.621i)11-s + (0.465 + 0.806i)12-s + (0.0262 + 0.00954i)13-s + (−0.126 + 0.716i)14-s + (−0.234 + 0.0855i)16-s + (−1.01 − 0.849i)17-s + 1.74·18-s + (0.271 + 0.962i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.75811 + 0.888122i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.75811 + 0.888122i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.766 - 0.642i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (-1.18 - 4.19i)T \) |
good | 3 | \( 1 + (-3.03 + 1.10i)T + (2.29 - 1.92i)T^{2} \) |
| 7 | \( 1 + (-1.36 - 2.35i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.18 - 2.05i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.0945 - 0.0344i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (4.17 + 3.50i)T + (2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (0.235 + 1.33i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (1.61 - 1.35i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (5.26 + 9.11i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 1.18T + 37T^{2} \) |
| 41 | \( 1 + (-3.30 + 1.20i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-1.22 + 6.96i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (6.13 - 5.14i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (-1.94 - 11.0i)T + (-49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (9.13 + 7.66i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (0.0303 + 0.172i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-4.17 + 3.50i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.240 + 1.36i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (-11.3 + 4.13i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (3.51 - 1.27i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-4.50 - 7.79i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (7.26 + 2.64i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-2.26 - 1.89i)T + (16.8 + 95.5i)T^{2} \) |
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
−9.571877093513995905645689300857, −9.140674467992033269117910015029, −8.220821627067200610077054438668, −7.68436283695006038001541529353, −6.96359757997434428965444506154, −5.86779914425796349203449960015, −4.67875691897416564507904381181, −3.69975562819799026184041951902, −2.55196499216346884142281972958, −1.95610188120990473741408970896,
1.60637105604830816388943734945, 2.73107728988361733658458566172, 3.60980201895130463466517039387, 4.32780415556304689852195374169, 5.16051472635002599242079896798, 6.74152689704890929971560817772, 7.65847308344578263536271118106, 8.456879324357710516758035917957, 9.128459342084706210311726898670, 9.999696861888304412449839843705