L(s) = 1 | − 2-s − 1.58i·3-s + 4-s − 3.32·5-s + 1.58i·6-s + 2.41i·7-s − 8-s + 0.483·9-s + 3.32·10-s + (−1.04 + 3.14i)11-s − 1.58i·12-s + 3.85·13-s − 2.41i·14-s + 5.26i·15-s + 16-s − 1.39i·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.915i·3-s + 0.5·4-s − 1.48·5-s + 0.647i·6-s + 0.914i·7-s − 0.353·8-s + 0.161·9-s + 1.05·10-s + (−0.314 + 0.949i)11-s − 0.457i·12-s + 1.07·13-s − 0.646i·14-s + 1.36i·15-s + 0.250·16-s − 0.337i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 - 0.345i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.938 - 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.746344 + 0.132874i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.746344 + 0.132874i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 11 | \( 1 + (1.04 - 3.14i)T \) |
| 19 | \( 1 + (-2.71 - 3.41i)T \) |
good | 3 | \( 1 + 1.58iT - 3T^{2} \) |
| 5 | \( 1 + 3.32T + 5T^{2} \) |
| 7 | \( 1 - 2.41iT - 7T^{2} \) |
| 13 | \( 1 - 3.85T + 13T^{2} \) |
| 17 | \( 1 + 1.39iT - 17T^{2} \) |
| 23 | \( 1 - 5.55T + 23T^{2} \) |
| 29 | \( 1 + 2.23T + 29T^{2} \) |
| 31 | \( 1 + 1.55iT - 31T^{2} \) |
| 37 | \( 1 - 5.70iT - 37T^{2} \) |
| 41 | \( 1 + 2.95T + 41T^{2} \) |
| 43 | \( 1 - 4.05iT - 43T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 - 12.9iT - 53T^{2} \) |
| 59 | \( 1 - 10.0iT - 59T^{2} \) |
| 61 | \( 1 - 10.3iT - 61T^{2} \) |
| 67 | \( 1 + 12.7iT - 67T^{2} \) |
| 71 | \( 1 - 7.99iT - 71T^{2} \) |
| 73 | \( 1 - 5.49iT - 73T^{2} \) |
| 79 | \( 1 + 3.59T + 79T^{2} \) |
| 83 | \( 1 - 1.78iT - 83T^{2} \) |
| 89 | \( 1 + 11.2iT - 89T^{2} \) |
| 97 | \( 1 + 4.62iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.45589007508691310852216407488, −10.41949601475226860048948962833, −9.203224158917698353918254235847, −8.347292275452378700603622715802, −7.55925315636334731786236286225, −7.03321508119472222057446871606, −5.77433624778691430361377935066, −4.25810879321570677501817729954, −2.83482269006274851332282380473, −1.27603212719977215711898238829,
0.74175404979455627038696214270, 3.40421306270928754850830862151, 3.91810145506553482724426769677, 5.19779152037793208245951230545, 6.81190860795510204624944034991, 7.58736444299034263166999890247, 8.497453735049580920300511640014, 9.246414661696288789909533790880, 10.51896372201704440976105469133, 10.94057550800698378528007654312