| L(s) = 1 | − 2.33i·2-s − 5.19·3-s + 10.5·4-s + 30.0·5-s + 12.1i·6-s − 41.7·7-s − 62.0i·8-s + 27·9-s − 70.2i·10-s − 117. i·11-s − 54.6·12-s + 118. i·13-s + 97.6i·14-s − 156.·15-s + 23.2·16-s + 263.·17-s + ⋯ |
| L(s) = 1 | − 0.584i·2-s − 0.577·3-s + 0.657·4-s + 1.20·5-s + 0.337i·6-s − 0.851·7-s − 0.969i·8-s + 0.333·9-s − 0.702i·10-s − 0.970i·11-s − 0.379·12-s + 0.700i·13-s + 0.498i·14-s − 0.693·15-s + 0.0906·16-s + 0.913·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00986 + 0.999i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.00986 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.104707025\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.104707025\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 5.19T \) |
| 59 | \( 1 + (34.3 - 3.48e3i)T \) |
| good | 2 | \( 1 + 2.33iT - 16T^{2} \) |
| 5 | \( 1 - 30.0T + 625T^{2} \) |
| 7 | \( 1 + 41.7T + 2.40e3T^{2} \) |
| 11 | \( 1 + 117. iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 118. iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 263.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 373.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 503. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 549.T + 7.07e5T^{2} \) |
| 31 | \( 1 + 450. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 1.43e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 1.44e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + 3.20e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 967. iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 1.66e3T + 7.89e6T^{2} \) |
| 61 | \( 1 - 1.71e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 5.06e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 6.15e3T + 2.54e7T^{2} \) |
| 73 | \( 1 - 4.17e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + 5.99e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 2.93e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 3.87e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 2.34e3iT - 8.85e7T^{2} \) |
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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
−11.72845617694548920006433341247, −10.75202941129290845532505835401, −9.953359739628721100411915762993, −9.187785079630040215202973552977, −7.31804260646091393653174709496, −6.23701687808694142468759816272, −5.62681773665568418840495778779, −3.60798355629115272383810948675, −2.30836344071872843655135661700, −0.870416092807677147748738602970,
1.52336710763147166432009793764, 3.03747549077897921450493634830, 5.24586761250215963824776413652, 5.90285949394732116276477343842, 6.86419734581886923533204962578, 7.80060090773495715455382747654, 9.645926101817594221761956460082, 9.995834527709638076297608763791, 11.27097540633265056810821627288, 12.33830999172682468630970489674
