| L(s) = 1 | + (−2.81 + 2.61i)2-s + (3.09 − 0.466i)3-s + (0.505 − 6.74i)4-s + (−3.96 + 10.0i)5-s + (−7.50 + 9.40i)6-s + (11.1 + 14.8i)7-s + (−2.96 − 3.71i)8-s + (−16.4 + 5.06i)9-s + (−15.2 − 38.8i)10-s + (−54.5 − 16.8i)11-s + (−1.58 − 21.1i)12-s + (14.0 + 61.6i)13-s + (−70.0 − 12.6i)14-s + (−7.56 + 33.1i)15-s + (71.5 + 10.7i)16-s + (54.5 − 37.2i)17-s + ⋯ |
| L(s) = 1 | + (−0.995 + 0.923i)2-s + (0.595 − 0.0898i)3-s + (0.0631 − 0.842i)4-s + (−0.354 + 0.903i)5-s + (−0.510 + 0.640i)6-s + (0.600 + 0.799i)7-s + (−0.131 − 0.164i)8-s + (−0.608 + 0.187i)9-s + (−0.481 − 1.22i)10-s + (−1.49 − 0.460i)11-s + (−0.0380 − 0.508i)12-s + (0.300 + 1.31i)13-s + (−1.33 − 0.241i)14-s + (−0.130 + 0.570i)15-s + (1.11 + 0.168i)16-s + (0.778 − 0.531i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.177176 + 0.750613i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.177176 + 0.750613i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-11.1 - 14.8i)T \) |
| good | 2 | \( 1 + (2.81 - 2.61i)T + (0.597 - 7.97i)T^{2} \) |
| 3 | \( 1 + (-3.09 + 0.466i)T + (25.8 - 7.95i)T^{2} \) |
| 5 | \( 1 + (3.96 - 10.0i)T + (-91.6 - 85.0i)T^{2} \) |
| 11 | \( 1 + (54.5 + 16.8i)T + (1.09e3 + 749. i)T^{2} \) |
| 13 | \( 1 + (-14.0 - 61.6i)T + (-1.97e3 + 953. i)T^{2} \) |
| 17 | \( 1 + (-54.5 + 37.2i)T + (1.79e3 - 4.57e3i)T^{2} \) |
| 19 | \( 1 + (-20.4 - 35.4i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-43.9 - 29.9i)T + (4.44e3 + 1.13e4i)T^{2} \) |
| 29 | \( 1 + (-216. - 104. i)T + (1.52e4 + 1.90e4i)T^{2} \) |
| 31 | \( 1 + (-96.9 + 167. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (13.6 + 182. i)T + (-5.00e4 + 7.54e3i)T^{2} \) |
| 41 | \( 1 + (71.1 + 89.2i)T + (-1.53e4 + 6.71e4i)T^{2} \) |
| 43 | \( 1 + (-64.4 + 80.7i)T + (-1.76e4 - 7.75e4i)T^{2} \) |
| 47 | \( 1 + (-157. + 145. i)T + (7.75e3 - 1.03e5i)T^{2} \) |
| 53 | \( 1 + (36.2 - 483. i)T + (-1.47e5 - 2.21e4i)T^{2} \) |
| 59 | \( 1 + (-175. - 447. i)T + (-1.50e5 + 1.39e5i)T^{2} \) |
| 61 | \( 1 + (-24.1 - 322. i)T + (-2.24e5 + 3.38e4i)T^{2} \) |
| 67 | \( 1 + (-231. + 400. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + (361. - 174. i)T + (2.23e5 - 2.79e5i)T^{2} \) |
| 73 | \( 1 + (664. + 616. i)T + (2.90e4 + 3.87e5i)T^{2} \) |
| 79 | \( 1 + (-448. - 776. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-163. + 717. i)T + (-5.15e5 - 2.48e5i)T^{2} \) |
| 89 | \( 1 + (-613. + 189. i)T + (5.82e5 - 3.97e5i)T^{2} \) |
| 97 | \( 1 + 757.T + 9.12e5T^{2} \) |
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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
−15.65383628482371997545871436666, −14.74359191545580918685784488613, −13.82355243079485482267123181289, −11.91231542416807998931070234671, −10.65623140028190941020083635870, −9.105418363853987575900681839835, −8.182149774335310302888047325439, −7.29902224536543359241274860213, −5.69055645475786769211829118630, −2.87882708185478696294269568621,
0.77617105870005371960383911273, 2.94958895095776351548345583364, 5.10509540132035638215139983857, 8.134334974268074340994752628378, 8.268118181693088510141822631239, 9.959883487947050698647292828439, 10.76059892869699899339638914428, 12.11496831674629463537388204621, 13.22748982822626248581304611831, 14.65343809267970564279711112098
