| L(s) = 1 | + (−0.841 + 0.540i)2-s + (0.0390 − 0.271i)3-s + (0.415 − 0.909i)4-s + (−0.959 + 0.281i)5-s + (0.113 + 0.249i)6-s + (−0.365 − 0.421i)7-s + (0.142 + 0.989i)8-s + (2.80 + 0.823i)9-s + (0.654 − 0.755i)10-s + (4.26 + 2.74i)11-s + (−0.230 − 0.148i)12-s + (2.67 − 3.08i)13-s + (0.534 + 0.157i)14-s + (0.0390 + 0.271i)15-s + (−0.654 − 0.755i)16-s + (−0.242 − 0.530i)17-s + ⋯ |
| L(s) = 1 | + (−0.594 + 0.382i)2-s + (0.0225 − 0.156i)3-s + (0.207 − 0.454i)4-s + (−0.429 + 0.125i)5-s + (0.0465 + 0.101i)6-s + (−0.137 − 0.159i)7-s + (0.0503 + 0.349i)8-s + (0.935 + 0.274i)9-s + (0.207 − 0.238i)10-s + (1.28 + 0.826i)11-s + (−0.0666 − 0.0428i)12-s + (0.740 − 0.855i)13-s + (0.142 + 0.0419i)14-s + (0.0100 + 0.0701i)15-s + (−0.163 − 0.188i)16-s + (−0.0588 − 0.128i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.372i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 - 0.372i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.966572 + 0.186970i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.966572 + 0.186970i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.841 - 0.540i)T \) |
| 5 | \( 1 + (0.959 - 0.281i)T \) |
| 23 | \( 1 + (-4.35 - 2.01i)T \) |
| good | 3 | \( 1 + (-0.0390 + 0.271i)T + (-2.87 - 0.845i)T^{2} \) |
| 7 | \( 1 + (0.365 + 0.421i)T + (-0.996 + 6.92i)T^{2} \) |
| 11 | \( 1 + (-4.26 - 2.74i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (-2.67 + 3.08i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (0.242 + 0.530i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (0.100 - 0.220i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (-0.883 - 1.93i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (-0.336 - 2.34i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (9.46 + 2.77i)T + (31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (-0.750 + 0.220i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (0.472 - 3.28i)T + (-41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 53 | \( 1 + (-1.08 - 1.25i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (-1.22 + 1.41i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (1.90 + 13.2i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (5.19 - 3.34i)T + (27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (-2.45 + 1.57i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (-3.30 + 7.22i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (9.00 - 10.3i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (3.65 + 1.07i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (2.21 - 15.3i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-7.91 + 2.32i)T + (81.6 - 52.4i)T^{2} \) |
| 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
−12.25264055244277154625060868476, −11.14705270744063675086396384737, −10.23675556161342901979415088614, −9.320153615487449108689123272905, −8.260495098420203327776337141408, −7.18986107738482313466093669084, −6.58376083991480649353770473307, −4.98122177688975546149929681781, −3.61085440815406198212439151492, −1.44850953080587453300354988779,
1.35022574500771262044869774451, 3.43086308162199840809946279215, 4.38177345671830588095266568878, 6.29746626253964667780239570182, 7.13477318519584192619020251330, 8.598204685548601913146655164177, 9.089316748526310332434150323117, 10.18852484399774199361500476807, 11.27226268203845724319682581418, 11.88159476825059535757235152074
