| L(s) = 1 | + (1.14 − 0.735i)2-s + (−2.21 + 15.4i)3-s + (−25.8 + 56.5i)4-s + (48.1 + 163. i)5-s + (8.80 + 19.2i)6-s + (−158. + 137. i)7-s + (24.4 + 169. i)8-s + (−233. − 68.4i)9-s + (175. + 152. i)10-s + (−96.8 + 150. i)11-s + (−815. − 523. i)12-s + (1.49e3 − 1.72e3i)13-s + (−80.1 + 273. i)14-s + (−2.63e3 + 379. i)15-s + (−2.45e3 − 2.82e3i)16-s + (1.46e3 − 670. i)17-s + ⋯ |
| L(s) = 1 | + (0.143 − 0.0919i)2-s + (−0.0821 + 0.571i)3-s + (−0.403 + 0.883i)4-s + (0.385 + 1.31i)5-s + (0.0407 + 0.0892i)6-s + (−0.460 + 0.399i)7-s + (0.0476 + 0.331i)8-s + (−0.319 − 0.0939i)9-s + (0.175 + 0.152i)10-s + (−0.0727 + 0.113i)11-s + (−0.471 − 0.303i)12-s + (0.680 − 0.785i)13-s + (−0.0292 + 0.0994i)14-s + (−0.781 + 0.112i)15-s + (−0.598 − 0.690i)16-s + (0.298 − 0.136i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.122i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.992 + 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.0752674 - 1.22786i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0752674 - 1.22786i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.21 - 15.4i)T \) |
| 23 | \( 1 + (1.04e3 - 1.21e4i)T \) |
| good | 2 | \( 1 + (-1.14 + 0.735i)T + (26.5 - 58.2i)T^{2} \) |
| 5 | \( 1 + (-48.1 - 163. i)T + (-1.31e4 + 8.44e3i)T^{2} \) |
| 7 | \( 1 + (158. - 137. i)T + (1.67e4 - 1.16e5i)T^{2} \) |
| 11 | \( 1 + (96.8 - 150. i)T + (-7.35e5 - 1.61e6i)T^{2} \) |
| 13 | \( 1 + (-1.49e3 + 1.72e3i)T + (-6.86e5 - 4.77e6i)T^{2} \) |
| 17 | \( 1 + (-1.46e3 + 670. i)T + (1.58e7 - 1.82e7i)T^{2} \) |
| 19 | \( 1 + (2.17e3 + 995. i)T + (3.08e7 + 3.55e7i)T^{2} \) |
| 29 | \( 1 + (6.76e3 + 1.48e4i)T + (-3.89e8 + 4.49e8i)T^{2} \) |
| 31 | \( 1 + (-5.91e3 - 4.11e4i)T + (-8.51e8 + 2.50e8i)T^{2} \) |
| 37 | \( 1 + (-6.77e3 + 2.30e4i)T + (-2.15e9 - 1.38e9i)T^{2} \) |
| 41 | \( 1 + (7.36e4 - 2.16e4i)T + (3.99e9 - 2.56e9i)T^{2} \) |
| 43 | \( 1 + (6.39e4 + 9.19e3i)T + (6.06e9 + 1.78e9i)T^{2} \) |
| 47 | \( 1 - 1.09e5T + 1.07e10T^{2} \) |
| 53 | \( 1 + (-1.39e5 + 1.20e5i)T + (3.15e9 - 2.19e10i)T^{2} \) |
| 59 | \( 1 + (-9.42e4 + 1.08e5i)T + (-6.00e9 - 4.17e10i)T^{2} \) |
| 61 | \( 1 + (5.52e4 - 7.94e3i)T + (4.94e10 - 1.45e10i)T^{2} \) |
| 67 | \( 1 + (-1.12e5 - 1.75e5i)T + (-3.75e10 + 8.22e10i)T^{2} \) |
| 71 | \( 1 + (5.37e4 - 3.45e4i)T + (5.32e10 - 1.16e11i)T^{2} \) |
| 73 | \( 1 + (3.05e5 - 6.69e5i)T + (-9.91e10 - 1.14e11i)T^{2} \) |
| 79 | \( 1 + (-3.12e5 - 2.70e5i)T + (3.45e10 + 2.40e11i)T^{2} \) |
| 83 | \( 1 + (-3.57e4 + 1.21e5i)T + (-2.75e11 - 1.76e11i)T^{2} \) |
| 89 | \( 1 + (3.67e5 + 5.28e4i)T + (4.76e11 + 1.40e11i)T^{2} \) |
| 97 | \( 1 + (2.18e4 + 7.43e4i)T + (-7.00e11 + 4.50e11i)T^{2} \) |
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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
−14.03186337506223938319754543801, −13.08289768447652813320955557531, −11.81091062057055956171540557390, −10.69708054856620216591601762753, −9.667566397918711141491497075521, −8.357704989737855056078088062816, −6.92740481292641007738292664046, −5.51635103767122023591391692821, −3.67015827982461723482363204359, −2.75929321387471990976344801212,
0.47767477730747780076238689310, 1.62011632638428245971662186947, 4.26897575604768066090237249146, 5.56385689057385917414381664209, 6.61589809906815679790142300227, 8.456325842936737357939286282865, 9.367474171342707605179011520692, 10.56485749309866341089712002285, 12.09626266139648623955078399769, 13.25272079163759959755243502542
