| L(s) = 1 | + (−43.5 − 75.4i)2-s + (5.62e3 − 9.74e3i)3-s + (6.17e4 − 1.06e5i)4-s + (−4.72e5 − 8.18e5i)5-s − 9.80e5·6-s + (1.52e7 − 6.74e5i)7-s − 2.21e7·8-s + (1.26e6 + 2.19e6i)9-s + (−4.11e7 + 7.13e7i)10-s + (4.92e7 − 8.53e7i)11-s + (−6.94e8 − 1.20e9i)12-s − 1.90e9·13-s + (−7.15e8 − 1.12e9i)14-s − 1.06e10·15-s + (−7.12e9 − 1.23e10i)16-s + (−6.75e9 + 1.17e10i)17-s + ⋯ |
| L(s) = 1 | + (−0.120 − 0.208i)2-s + (0.495 − 0.857i)3-s + (0.471 − 0.815i)4-s + (−0.541 − 0.937i)5-s − 0.238·6-s + (0.999 − 0.0442i)7-s − 0.467·8-s + (0.00980 + 0.0169i)9-s + (−0.130 + 0.225i)10-s + (0.0693 − 0.120i)11-s + (−0.466 − 0.807i)12-s − 0.649·13-s + (−0.129 − 0.202i)14-s − 1.07·15-s + (−0.414 − 0.718i)16-s + (−0.234 + 0.406i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.875 + 0.483i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (-0.875 + 0.483i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(0.494878 - 1.92025i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.494878 - 1.92025i\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-1.52e7 + 6.74e5i)T \) |
| good | 2 | \( 1 + (43.5 + 75.4i)T + (-6.55e4 + 1.13e5i)T^{2} \) |
| 3 | \( 1 + (-5.62e3 + 9.74e3i)T + (-6.45e7 - 1.11e8i)T^{2} \) |
| 5 | \( 1 + (4.72e5 + 8.18e5i)T + (-3.81e11 + 6.60e11i)T^{2} \) |
| 11 | \( 1 + (-4.92e7 + 8.53e7i)T + (-2.52e17 - 4.37e17i)T^{2} \) |
| 13 | \( 1 + 1.90e9T + 8.65e18T^{2} \) |
| 17 | \( 1 + (6.75e9 - 1.17e10i)T + (-4.13e20 - 7.16e20i)T^{2} \) |
| 19 | \( 1 + (4.54e10 + 7.87e10i)T + (-2.74e21 + 4.74e21i)T^{2} \) |
| 23 | \( 1 + (-2.91e11 - 5.05e11i)T + (-7.05e22 + 1.22e23i)T^{2} \) |
| 29 | \( 1 - 1.73e12T + 7.25e24T^{2} \) |
| 31 | \( 1 + (2.04e12 - 3.54e12i)T + (-1.12e25 - 1.95e25i)T^{2} \) |
| 37 | \( 1 + (1.95e13 + 3.38e13i)T + (-2.28e26 + 3.95e26i)T^{2} \) |
| 41 | \( 1 + 1.74e13T + 2.61e27T^{2} \) |
| 43 | \( 1 - 1.09e14T + 5.87e27T^{2} \) |
| 47 | \( 1 + (-9.14e13 - 1.58e14i)T + (-1.33e28 + 2.30e28i)T^{2} \) |
| 53 | \( 1 + (-1.81e14 + 3.13e14i)T + (-1.02e29 - 1.77e29i)T^{2} \) |
| 59 | \( 1 + (-8.91e14 + 1.54e15i)T + (-6.35e29 - 1.10e30i)T^{2} \) |
| 61 | \( 1 + (4.52e14 + 7.83e14i)T + (-1.12e30 + 1.94e30i)T^{2} \) |
| 67 | \( 1 + (3.65e14 - 6.32e14i)T + (-5.52e30 - 9.56e30i)T^{2} \) |
| 71 | \( 1 - 2.59e15T + 2.96e31T^{2} \) |
| 73 | \( 1 + (-3.81e15 + 6.60e15i)T + (-2.37e31 - 4.11e31i)T^{2} \) |
| 79 | \( 1 + (-3.19e15 - 5.52e15i)T + (-9.09e31 + 1.57e32i)T^{2} \) |
| 83 | \( 1 + 3.88e16T + 4.21e32T^{2} \) |
| 89 | \( 1 + (-2.80e15 - 4.85e15i)T + (-6.89e32 + 1.19e33i)T^{2} \) |
| 97 | \( 1 - 4.46e16T + 5.95e33T^{2} \) |
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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
−17.58359621353591920430977719977, −15.65987995321614940319554845291, −14.20343692324983421124736515889, −12.54039132820394582798698862693, −11.03443824216113287698549353389, −8.795115377689558425890318579711, −7.28518866021160396986104047379, −4.97182429234356355866855118322, −2.07057121376433568657266675020, −0.858430966126612955493526542994,
2.70201720236116938303659507951, 4.18715009185534717347457480973, 7.07285577334456846106848994014, 8.525665161421001114330462343149, 10.59348287282845771192458261268, 12.04527932321259890336012035306, 14.60741702343615236710050373161, 15.35742039142702787055682820640, 16.93622912236685700266983471334, 18.51327286278800007106797636944
