L(s) = 1 | − 0.363i·3-s + (−1.75 − 1.38i)5-s + i·7-s + 2.86·9-s − 5.14·11-s − 4.64i·13-s + (−0.504 + 0.636i)15-s − 3.86i·17-s − 0.778·19-s + 0.363·21-s − 5.00i·23-s + (1.14 + 4.86i)25-s − 2.13i·27-s − 9.42·29-s − 4.72·31-s + ⋯ |
L(s) = 1 | − 0.209i·3-s + (−0.783 − 0.621i)5-s + 0.377i·7-s + 0.955·9-s − 1.55·11-s − 1.28i·13-s + (−0.130 + 0.164i)15-s − 0.938i·17-s − 0.178·19-s + 0.0792·21-s − 1.04i·23-s + (0.228 + 0.973i)25-s − 0.410i·27-s − 1.74·29-s − 0.848·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.621 + 0.783i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.621 + 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.349946 - 0.723937i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.349946 - 0.723937i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.75 + 1.38i)T \) |
| 7 | \( 1 - iT \) |
good | 3 | \( 1 + 0.363iT - 3T^{2} \) |
| 11 | \( 1 + 5.14T + 11T^{2} \) |
| 13 | \( 1 + 4.64iT - 13T^{2} \) |
| 17 | \( 1 + 3.86iT - 17T^{2} \) |
| 19 | \( 1 + 0.778T + 19T^{2} \) |
| 23 | \( 1 + 5.00iT - 23T^{2} \) |
| 29 | \( 1 + 9.42T + 29T^{2} \) |
| 31 | \( 1 + 4.72T + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 + 1.00T + 41T^{2} \) |
| 43 | \( 1 - 7.00iT - 43T^{2} \) |
| 47 | \( 1 + 11.4iT - 47T^{2} \) |
| 53 | \( 1 - 7.55iT - 53T^{2} \) |
| 59 | \( 1 - 12.5T + 59T^{2} \) |
| 61 | \( 1 - 11.5T + 61T^{2} \) |
| 67 | \( 1 + 11.7iT - 67T^{2} \) |
| 71 | \( 1 + 2.72T + 71T^{2} \) |
| 73 | \( 1 - 5.00iT - 73T^{2} \) |
| 79 | \( 1 - 5.68T + 79T^{2} \) |
| 83 | \( 1 - 4.67iT - 83T^{2} \) |
| 89 | \( 1 - 2.82T + 89T^{2} \) |
| 97 | \( 1 - 1.58iT - 97T^{2} \) |
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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
−10.51122538332543913015797817205, −9.589074022659839418017664758910, −8.512168878131937771699206445227, −7.74384845516185904536577837324, −7.16201970589236495679756126986, −5.56398925708776668727302718891, −4.94506077533364295543517149083, −3.68748847284982373605930481326, −2.35820578998841619374724966566, −0.44648047525435250959307692164,
1.96853018671105723091098124064, 3.55264762165505726585407964457, 4.26698219879601631519074926766, 5.46359701149597170961347255797, 6.84027301768124168392837837516, 7.42974978681483996921058270604, 8.248203073897564330384074295322, 9.490734512353544634730230054291, 10.32441230253612989618182669878, 10.98244029861962307417181766209