L(s) = 1 | + (0.0324 + 1.41i)2-s + (2.32 − 2.32i)3-s + (−1.99 + 0.0918i)4-s + (−2.23 − 0.0191i)5-s + (3.36 + 3.21i)6-s + (1.97 − 1.97i)7-s + (−0.194 − 2.82i)8-s − 7.85i·9-s + (−0.0455 − 3.16i)10-s + 2.76·11-s + (−4.44 + 4.86i)12-s + (0.543 + 3.56i)13-s + (2.85 + 2.72i)14-s + (−5.25 + 5.16i)15-s + (3.98 − 0.367i)16-s + (−1.79 + 1.79i)17-s + ⋯ |
L(s) = 1 | + (0.0229 + 0.999i)2-s + (1.34 − 1.34i)3-s + (−0.998 + 0.0459i)4-s + (−0.999 − 0.00856i)5-s + (1.37 + 1.31i)6-s + (0.745 − 0.745i)7-s + (−0.0688 − 0.997i)8-s − 2.61i·9-s + (−0.0144 − 0.999i)10-s + 0.833·11-s + (−1.28 + 1.40i)12-s + (0.150 + 0.988i)13-s + (0.761 + 0.727i)14-s + (−1.35 + 1.33i)15-s + (0.995 − 0.0917i)16-s + (−0.436 + 0.436i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 + 0.356i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.934 + 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.57628 - 0.290772i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.57628 - 0.290772i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.0324 - 1.41i)T \) |
| 5 | \( 1 + (2.23 + 0.0191i)T \) |
| 13 | \( 1 + (-0.543 - 3.56i)T \) |
good | 3 | \( 1 + (-2.32 + 2.32i)T - 3iT^{2} \) |
| 7 | \( 1 + (-1.97 + 1.97i)T - 7iT^{2} \) |
| 11 | \( 1 - 2.76T + 11T^{2} \) |
| 17 | \( 1 + (1.79 - 1.79i)T - 17iT^{2} \) |
| 19 | \( 1 + 2.06iT - 19T^{2} \) |
| 23 | \( 1 + (1.07 - 1.07i)T - 23iT^{2} \) |
| 29 | \( 1 - 7.19iT - 29T^{2} \) |
| 31 | \( 1 + 2.08T + 31T^{2} \) |
| 37 | \( 1 + (-2.10 - 2.10i)T + 37iT^{2} \) |
| 41 | \( 1 - 2.07iT - 41T^{2} \) |
| 43 | \( 1 + (1.37 - 1.37i)T - 43iT^{2} \) |
| 47 | \( 1 + (3.03 - 3.03i)T - 47iT^{2} \) |
| 53 | \( 1 + (4.80 + 4.80i)T + 53iT^{2} \) |
| 59 | \( 1 - 9.65iT - 59T^{2} \) |
| 61 | \( 1 - 6.83T + 61T^{2} \) |
| 67 | \( 1 + (-10.5 + 10.5i)T - 67iT^{2} \) |
| 71 | \( 1 + 3.76T + 71T^{2} \) |
| 73 | \( 1 + (0.228 - 0.228i)T - 73iT^{2} \) |
| 79 | \( 1 - 3.36T + 79T^{2} \) |
| 83 | \( 1 + (-2.38 - 2.38i)T + 83iT^{2} \) |
| 89 | \( 1 + 3.39T + 89T^{2} \) |
| 97 | \( 1 + (5.33 + 5.33i)T + 97iT^{2} \) |
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
−12.26895077914676003169025203775, −11.24008844844945808407221085096, −9.363227972804669491485228511579, −8.606064969570401817724275040153, −7.934434081619894046262047122753, −7.08972531447285612144671275692, −6.58359070095098914419987994361, −4.42844138703313869180648893819, −3.52937301380399860383654970160, −1.34817122601768614345962684741,
2.34292580241860278893345742942, 3.50732163849079474781208376117, 4.27435519553649990878734499354, 5.26438877404942042885127280994, 7.902569257309609228060788015845, 8.423987835295432503971922172528, 9.202412533835968128025506322819, 10.10936885109408847914084601424, 11.07771940621783617309180444061, 11.74392130030287719635783202230