| L(s) = 1 | − 11.2i·2-s + 19.1i·3-s − 94.0·4-s + 215.·6-s − 70.9i·7-s + 696. i·8-s − 125.·9-s − 161.·11-s − 1.80e3i·12-s + 169i·13-s − 796.·14-s + 4.80e3·16-s + 121. i·17-s + 1.40e3i·18-s + 3.11e3·19-s + ⋯ |
| L(s) = 1 | − 1.98i·2-s + 1.23i·3-s − 2.93·4-s + 2.44·6-s − 0.547i·7-s + 3.84i·8-s − 0.515·9-s − 0.401·11-s − 3.61i·12-s + 0.277i·13-s − 1.08·14-s + 4.69·16-s + 0.101i·17-s + 1.02i·18-s + 1.97·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.2319695603\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2319695603\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 - 169iT \) |
| good | 2 | \( 1 + 11.2iT - 32T^{2} \) |
| 3 | \( 1 - 19.1iT - 243T^{2} \) |
| 7 | \( 1 + 70.9iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 161.T + 1.61e5T^{2} \) |
| 17 | \( 1 - 121. iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 3.11e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.09e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 3.17e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.51e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 6.99e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 1.95e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.33e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 7.63e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 2.77e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 3.30e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.31e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.45e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 1.66e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.25e3iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 3.12e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.51e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 634.T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.56e5iT - 8.58e9T^{2} \) |
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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
−10.37495893611789225998182538609, −9.550810987197325993177104739603, −8.988462148402855868857561770484, −7.69511333762359243813683008810, −5.40658069393873363384542486104, −4.61576196491944571205065296332, −3.74909990046918467073142160851, −2.92836787524459980163867653520, −1.43476455682323609521995315516, −0.06993881135606415513468755086,
1.27054786711173400915979774472, 3.43806397749804201964257203973, 5.15779066931417445457433534902, 5.69245967362803549093878154765, 6.80735299296492054612977253762, 7.48204204050828274333532016217, 8.056722694638745836980227475367, 9.110731259413024074958497301342, 9.941441555435790639911985264019, 11.83221017044450593585265106091
