| L(s) = 1 | − 6.32i·2-s + (24.9 − 10.1i)3-s + 23.9·4-s + 138. i·5-s + (−64.5 − 158. i)6-s − 112.·7-s − 556. i·8-s + (520. − 509. i)9-s + 873.·10-s − 88.8i·11-s + (599. − 244. i)12-s + 3.82e3·13-s + 713. i·14-s + (1.40e3 + 3.45e3i)15-s − 1.98e3·16-s − 3.73e3i·17-s + ⋯ |
| L(s) = 1 | − 0.790i·2-s + (0.925 − 0.377i)3-s + 0.374·4-s + 1.10i·5-s + (−0.298 − 0.732i)6-s − 0.329·7-s − 1.08i·8-s + (0.714 − 0.699i)9-s + 0.873·10-s − 0.0667i·11-s + (0.347 − 0.141i)12-s + 1.74·13-s + 0.260i·14-s + (0.417 + 1.02i)15-s − 0.484·16-s − 0.759i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.377 + 0.925i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.377 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.53438 - 1.70328i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.53438 - 1.70328i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-24.9 + 10.1i)T \) |
| 23 | \( 1 + 2.53e3iT \) |
| good | 2 | \( 1 + 6.32iT - 64T^{2} \) |
| 5 | \( 1 - 138. iT - 1.56e4T^{2} \) |
| 7 | \( 1 + 112.T + 1.17e5T^{2} \) |
| 11 | \( 1 + 88.8iT - 1.77e6T^{2} \) |
| 13 | \( 1 - 3.82e3T + 4.82e6T^{2} \) |
| 17 | \( 1 + 3.73e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 4.06e3T + 4.70e7T^{2} \) |
| 29 | \( 1 - 2.35e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 - 2.26e4T + 8.87e8T^{2} \) |
| 37 | \( 1 + 9.32e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + 5.44e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 1.08e5T + 6.32e9T^{2} \) |
| 47 | \( 1 - 9.85e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 2.68e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 2.04e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 3.39e5T + 5.15e10T^{2} \) |
| 67 | \( 1 + 2.21e5T + 9.04e10T^{2} \) |
| 71 | \( 1 + 1.44e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 9.52e4T + 1.51e11T^{2} \) |
| 79 | \( 1 + 2.75e5T + 2.43e11T^{2} \) |
| 83 | \( 1 - 8.39e4iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 1.57e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 2.13e4T + 8.32e11T^{2} \) |
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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
−13.36136688217560461150773333614, −12.13183948463735375171988595515, −11.01166696486971321405363970426, −10.11823684996684933439643372858, −8.801098836612999043535741579507, −7.23873135736701148189929825046, −6.42313294935464112734943465580, −3.55667159704218428417888237826, −2.85904394736212758973019962290, −1.32720986616021236205822425580,
1.61021348620242982773265487528, 3.55804602829166072956216866835, 5.16282671050523488902974390244, 6.57839123713009539601304998150, 8.186660673550549443287247870081, 8.636572024915572766901835274645, 10.06827438446795673645569042615, 11.50367724164326827279229921246, 13.00533519469397445712315667132, 13.79730498987918969740567802158
