L(s) = 1 | + (9.11 + 10.8i)3-s + (31.4 + 11.4i)5-s + (39.9 − 69.2i)7-s + (−20.8 + 118. i)9-s + (−10.1 − 17.6i)11-s + (−101. + 120. i)13-s + (162. + 445. i)15-s + (−64.2 − 364. i)17-s + (−169. + 318. i)19-s + (1.11e3 − 196. i)21-s + (−341. + 124. i)23-s + (377. + 316. i)25-s + (−480. + 277. i)27-s + (−681. − 120. i)29-s + (889. + 513. i)31-s + ⋯ |
L(s) = 1 | + (1.01 + 1.20i)3-s + (1.25 + 0.457i)5-s + (0.816 − 1.41i)7-s + (−0.257 + 1.46i)9-s + (−0.0839 − 0.145i)11-s + (−0.598 + 0.712i)13-s + (0.720 + 1.98i)15-s + (−0.222 − 1.26i)17-s + (−0.470 + 0.882i)19-s + (2.53 − 0.446i)21-s + (−0.644 + 0.234i)23-s + (0.604 + 0.507i)25-s + (−0.658 + 0.380i)27-s + (−0.809 − 0.142i)29-s + (0.925 + 0.534i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.598 - 0.801i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.598 - 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(2.45084 + 1.22868i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.45084 + 1.22868i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (169. - 318. i)T \) |
good | 3 | \( 1 + (-9.11 - 10.8i)T + (-14.0 + 79.7i)T^{2} \) |
| 5 | \( 1 + (-31.4 - 11.4i)T + (478. + 401. i)T^{2} \) |
| 7 | \( 1 + (-39.9 + 69.2i)T + (-1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + (10.1 + 17.6i)T + (-7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 + (101. - 120. i)T + (-4.95e3 - 2.81e4i)T^{2} \) |
| 17 | \( 1 + (64.2 + 364. i)T + (-7.84e4 + 2.85e4i)T^{2} \) |
| 23 | \( 1 + (341. - 124. i)T + (2.14e5 - 1.79e5i)T^{2} \) |
| 29 | \( 1 + (681. + 120. i)T + (6.64e5 + 2.41e5i)T^{2} \) |
| 31 | \( 1 + (-889. - 513. i)T + (4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 + 694. iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (1.06e3 + 1.26e3i)T + (-4.90e5 + 2.78e6i)T^{2} \) |
| 43 | \( 1 + (3.22e3 + 1.17e3i)T + (2.61e6 + 2.19e6i)T^{2} \) |
| 47 | \( 1 + (468. - 2.65e3i)T + (-4.58e6 - 1.66e6i)T^{2} \) |
| 53 | \( 1 + (767. + 2.10e3i)T + (-6.04e6 + 5.07e6i)T^{2} \) |
| 59 | \( 1 + (-6.50e3 + 1.14e3i)T + (1.13e7 - 4.14e6i)T^{2} \) |
| 61 | \( 1 + (-5.81e3 + 2.11e3i)T + (1.06e7 - 8.89e6i)T^{2} \) |
| 67 | \( 1 + (-6.48e3 - 1.14e3i)T + (1.89e7 + 6.89e6i)T^{2} \) |
| 71 | \( 1 + (1.83e3 - 5.03e3i)T + (-1.94e7 - 1.63e7i)T^{2} \) |
| 73 | \( 1 + (2.41e3 - 2.02e3i)T + (4.93e6 - 2.79e7i)T^{2} \) |
| 79 | \( 1 + (1.72e3 + 2.05e3i)T + (-6.76e6 + 3.83e7i)T^{2} \) |
| 83 | \( 1 + (-1.76e3 + 3.04e3i)T + (-2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 + (6.59e3 - 7.86e3i)T + (-1.08e7 - 6.17e7i)T^{2} \) |
| 97 | \( 1 + (-1.16e4 + 2.06e3i)T + (8.31e7 - 3.02e7i)T^{2} \) |
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
−14.23736786403934034841288854600, −13.47144886757430926703583480626, −11.33495669957495651303149025385, −10.10271959920793707307840047503, −9.817101323763884780507571552130, −8.385466646173589122876818869706, −6.98627633768060188305407143472, −5.04243816536271856092416293247, −3.82090414032914191124833395104, −2.13034091103782880256289140931,
1.73268254301245776097501695169, 2.48277737359146327992311185291, 5.25644781719929393218915225784, 6.45073889021039175926911078416, 8.145142968526116911063142281455, 8.692405756723057768671551081130, 9.928104487196624990558119495465, 11.79546568818705746482398974288, 12.89690537952336264613680178927, 13.34483984066417046046403519408