| L(s) = 1 | + (1.53 − 0.986i)2-s + (−1.28 + 8.90i)3-s + (−11.9 + 26.0i)4-s + (85.2 − 25.0i)5-s + (6.82 + 14.9i)6-s + (−23.0 − 26.6i)7-s + (15.7 + 109. i)8-s + (−77.7 − 22.8i)9-s + (106. − 122. i)10-s + (289. + 186. i)11-s + (−217. − 139. i)12-s + (−504. + 581. i)13-s + (−61.7 − 18.1i)14-s + (113. + 791. i)15-s + (−468. − 540. i)16-s + (878. + 1.92e3i)17-s + ⋯ |
| L(s) = 1 | + (0.271 − 0.174i)2-s + (−0.0821 + 0.571i)3-s + (−0.372 + 0.814i)4-s + (1.52 − 0.447i)5-s + (0.0773 + 0.169i)6-s + (−0.178 − 0.205i)7-s + (0.0870 + 0.605i)8-s + (−0.319 − 0.0939i)9-s + (0.335 − 0.387i)10-s + (0.722 + 0.464i)11-s + (−0.435 − 0.279i)12-s + (−0.827 + 0.955i)13-s + (−0.0842 − 0.0247i)14-s + (0.130 + 0.908i)15-s + (−0.457 − 0.527i)16-s + (0.737 + 1.61i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.189 - 0.981i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.189 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.64484 + 1.35729i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.64484 + 1.35729i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.28 - 8.90i)T \) |
| 23 | \( 1 + (-1.07e3 + 2.29e3i)T \) |
| good | 2 | \( 1 + (-1.53 + 0.986i)T + (13.2 - 29.1i)T^{2} \) |
| 5 | \( 1 + (-85.2 + 25.0i)T + (2.62e3 - 1.68e3i)T^{2} \) |
| 7 | \( 1 + (23.0 + 26.6i)T + (-2.39e3 + 1.66e4i)T^{2} \) |
| 11 | \( 1 + (-289. - 186. i)T + (6.69e4 + 1.46e5i)T^{2} \) |
| 13 | \( 1 + (504. - 581. i)T + (-5.28e4 - 3.67e5i)T^{2} \) |
| 17 | \( 1 + (-878. - 1.92e3i)T + (-9.29e5 + 1.07e6i)T^{2} \) |
| 19 | \( 1 + (478. - 1.04e3i)T + (-1.62e6 - 1.87e6i)T^{2} \) |
| 29 | \( 1 + (1.05e3 + 2.30e3i)T + (-1.34e7 + 1.55e7i)T^{2} \) |
| 31 | \( 1 + (-1.05e3 - 7.36e3i)T + (-2.74e7 + 8.06e6i)T^{2} \) |
| 37 | \( 1 + (-674. - 198. i)T + (5.83e7 + 3.74e7i)T^{2} \) |
| 41 | \( 1 + (3.86e3 - 1.13e3i)T + (9.74e7 - 6.26e7i)T^{2} \) |
| 43 | \( 1 + (-1.62e3 + 1.13e4i)T + (-1.41e8 - 4.14e7i)T^{2} \) |
| 47 | \( 1 - 1.47e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + (1.64e4 + 1.90e4i)T + (-5.95e7 + 4.13e8i)T^{2} \) |
| 59 | \( 1 + (-2.55e4 + 2.94e4i)T + (-1.01e8 - 7.07e8i)T^{2} \) |
| 61 | \( 1 + (-3.11e3 - 2.16e4i)T + (-8.10e8 + 2.37e8i)T^{2} \) |
| 67 | \( 1 + (-4.06e4 + 2.61e4i)T + (5.60e8 - 1.22e9i)T^{2} \) |
| 71 | \( 1 + (-1.42e4 + 9.16e3i)T + (7.49e8 - 1.64e9i)T^{2} \) |
| 73 | \( 1 + (434. - 951. i)T + (-1.35e9 - 1.56e9i)T^{2} \) |
| 79 | \( 1 + (-4.48e4 + 5.17e4i)T + (-4.37e8 - 3.04e9i)T^{2} \) |
| 83 | \( 1 + (1.35e4 + 3.98e3i)T + (3.31e9 + 2.12e9i)T^{2} \) |
| 89 | \( 1 + (3.91e3 - 2.72e4i)T + (-5.35e9 - 1.57e9i)T^{2} \) |
| 97 | \( 1 + (1.41e5 - 4.16e4i)T + (7.22e9 - 4.64e9i)T^{2} \) |
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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
−14.02369355173290415760979348087, −12.81304356013600639291973443428, −12.11159786997542192007299084665, −10.38068037859784282296043804224, −9.488776247582519024339763514567, −8.500256883106751366056438711005, −6.58711642181557502717512120062, −5.13056429923460002696498153931, −3.89730509226258660649191813527, −1.98436442711491573515199288212,
0.952630292829287247308887246003, 2.66010346458840398917831090840, 5.24427741491662211993093882657, 6.01004546514713829070371239457, 7.17576264115827790900354275369, 9.271387766807202502666140153194, 9.864209017005701509216721704065, 11.20915719677381227568188750480, 12.78137057275085834987806493515, 13.72028820562207069568456020295
