L(s) = 1 | + (−1.07 − 0.922i)2-s + (−2.52 − 1.76i)3-s + (0.297 + 1.97i)4-s + (0.172 + 0.0150i)5-s + (1.07 + 4.21i)6-s + (3.01 + 1.74i)7-s + (1.50 − 2.39i)8-s + (2.21 + 6.08i)9-s + (−0.170 − 0.174i)10-s + (0.624 − 2.33i)11-s + (2.74 − 5.51i)12-s + (1.34 + 1.92i)13-s + (−1.62 − 4.64i)14-s + (−0.407 − 0.341i)15-s + (−3.82 + 1.17i)16-s + (5.11 + 1.86i)17-s + ⋯ |
L(s) = 1 | + (−0.757 − 0.652i)2-s + (−1.45 − 1.01i)3-s + (0.148 + 0.988i)4-s + (0.0769 + 0.00673i)5-s + (0.438 + 1.72i)6-s + (1.13 + 0.657i)7-s + (0.532 − 0.846i)8-s + (0.738 + 2.02i)9-s + (−0.0539 − 0.0552i)10-s + (0.188 − 0.702i)11-s + (0.791 − 1.59i)12-s + (0.373 + 0.533i)13-s + (−0.434 − 1.24i)14-s + (−0.105 − 0.0882i)15-s + (−0.955 + 0.294i)16-s + (1.24 + 0.451i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.218 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.218 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.512027 - 0.410244i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.512027 - 0.410244i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.07 + 0.922i)T \) |
| 19 | \( 1 + (4.29 + 0.741i)T \) |
good | 3 | \( 1 + (2.52 + 1.76i)T + (1.02 + 2.81i)T^{2} \) |
| 5 | \( 1 + (-0.172 - 0.0150i)T + (4.92 + 0.868i)T^{2} \) |
| 7 | \( 1 + (-3.01 - 1.74i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.624 + 2.33i)T + (-9.52 - 5.5i)T^{2} \) |
| 13 | \( 1 + (-1.34 - 1.92i)T + (-4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (-5.11 - 1.86i)T + (13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (-3.02 + 3.60i)T + (-3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-1.31 + 2.80i)T + (-18.6 - 22.2i)T^{2} \) |
| 31 | \( 1 + (-3.56 + 6.16i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-7.65 - 7.65i)T + 37iT^{2} \) |
| 41 | \( 1 + (-1.71 + 0.302i)T + (38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-4.19 - 0.367i)T + (42.3 + 7.46i)T^{2} \) |
| 47 | \( 1 + (1.06 - 0.388i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (0.862 + 9.85i)T + (-52.1 + 9.20i)T^{2} \) |
| 59 | \( 1 + (-4.13 - 8.87i)T + (-37.9 + 45.1i)T^{2} \) |
| 61 | \( 1 + (-4.04 + 0.354i)T + (60.0 - 10.5i)T^{2} \) |
| 67 | \( 1 + (-1.57 + 3.38i)T + (-43.0 - 51.3i)T^{2} \) |
| 71 | \( 1 + (3.37 + 4.02i)T + (-12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (-10.1 + 1.78i)T + (68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (-1.40 - 7.99i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-0.233 - 0.870i)T + (-71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (11.9 + 2.10i)T + (83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (13.5 + 4.94i)T + (74.3 + 62.3i)T^{2} \) |
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
−11.43351165934819285339285714796, −11.03445924941255871170717732674, −9.901069596364723858716418018640, −8.400393473478623156901195203825, −7.914744796629059996381731519229, −6.58739291668197253838742186672, −5.77805788952250548192744934535, −4.41411129633436809218668958748, −2.22279078806759732604496005992, −1.01609365943535487323314858281,
1.13058486895534614679783565429, 4.18987346701190520947818417604, 5.11987240353211369955246735891, 5.84871097323553393028556533505, 7.05523054607824213670564061174, 8.006304611022298324225588690978, 9.365834787031487378088562768385, 10.14415871353594252912290302554, 10.84553896440505164855650776569, 11.41332669886927713356329852569