| 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 | \( 1 + (-1.16e4 - 2.06e3i)T + (8.31e7 + 3.02e7i)T^{2} \) |
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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
−13.34483984066417046046403519408, −12.89690537952336264613680178927, −11.79546568818705746482398974288, −9.928104487196624990558119495465, −8.692405756723057768671551081130, −8.145142968526116911063142281455, −6.45073889021039175926911078416, −5.25644781719929393218915225784, −2.48277737359146327992311185291, −1.73268254301245776097501695169,
2.13034091103782880256289140931, 3.82090414032914191124833395104, 5.04243816536271856092416293247, 6.98627633768060188305407143472, 8.385466646173589122876818869706, 9.817101323763884780507571552130, 10.10271959920793707307840047503, 11.33495669957495651303149025385, 13.47144886757430926703583480626, 14.23736786403934034841288854600
