| L(s) = 1 | + (−4.71 + 4.37i)2-s + (17.4 − 2.62i)3-s + (0.700 − 9.35i)4-s + (26.8 − 68.3i)5-s + (−70.6 + 88.6i)6-s + (−49.2 − 119. i)7-s + (−90.7 − 113. i)8-s + (64.5 − 19.9i)9-s + (172. + 439. i)10-s + (−503. − 155. i)11-s + (−12.3 − 164. i)12-s + (−95.0 − 416. i)13-s + (756. + 349. i)14-s + (288. − 1.26e3i)15-s + (1.22e3 + 184. i)16-s + (1.37e3 − 939. i)17-s + ⋯ |
| L(s) = 1 | + (−0.833 + 0.773i)2-s + (1.11 − 0.168i)3-s + (0.0219 − 0.292i)4-s + (0.480 − 1.22i)5-s + (−0.801 + 1.00i)6-s + (−0.379 − 0.925i)7-s + (−0.501 − 0.628i)8-s + (0.265 − 0.0819i)9-s + (0.545 + 1.39i)10-s + (−1.25 − 0.387i)11-s + (−0.0247 − 0.330i)12-s + (−0.155 − 0.683i)13-s + (1.03 + 0.477i)14-s + (0.330 − 1.44i)15-s + (1.19 + 0.179i)16-s + (1.15 − 0.788i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.562 + 0.826i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.562 + 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.09922 - 0.581369i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.09922 - 0.581369i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (49.2 + 119. i)T \) |
| good | 2 | \( 1 + (4.71 - 4.37i)T + (2.39 - 31.9i)T^{2} \) |
| 3 | \( 1 + (-17.4 + 2.62i)T + (232. - 71.6i)T^{2} \) |
| 5 | \( 1 + (-26.8 + 68.3i)T + (-2.29e3 - 2.12e3i)T^{2} \) |
| 11 | \( 1 + (503. + 155. i)T + (1.33e5 + 9.07e4i)T^{2} \) |
| 13 | \( 1 + (95.0 + 416. i)T + (-3.34e5 + 1.61e5i)T^{2} \) |
| 17 | \( 1 + (-1.37e3 + 939. i)T + (5.18e5 - 1.32e6i)T^{2} \) |
| 19 | \( 1 + (-622. - 1.07e3i)T + (-1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + (-104. - 71.0i)T + (2.35e6 + 5.99e6i)T^{2} \) |
| 29 | \( 1 + (-2.68e3 - 1.29e3i)T + (1.27e7 + 1.60e7i)T^{2} \) |
| 31 | \( 1 + (1.61e3 - 2.80e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + (575. + 7.67e3i)T + (-6.85e7 + 1.03e7i)T^{2} \) |
| 41 | \( 1 + (-1.29e4 - 1.62e4i)T + (-2.57e7 + 1.12e8i)T^{2} \) |
| 43 | \( 1 + (-4.47e3 + 5.61e3i)T + (-3.27e7 - 1.43e8i)T^{2} \) |
| 47 | \( 1 + (-1.12e4 + 1.04e4i)T + (1.71e7 - 2.28e8i)T^{2} \) |
| 53 | \( 1 + (-2.04e3 + 2.72e4i)T + (-4.13e8 - 6.23e7i)T^{2} \) |
| 59 | \( 1 + (1.59e4 + 4.07e4i)T + (-5.24e8 + 4.86e8i)T^{2} \) |
| 61 | \( 1 + (-2.00e3 - 2.66e4i)T + (-8.35e8 + 1.25e8i)T^{2} \) |
| 67 | \( 1 + (2.82e4 - 4.88e4i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 + (-5.79e4 + 2.79e4i)T + (1.12e9 - 1.41e9i)T^{2} \) |
| 73 | \( 1 + (1.83e4 + 1.70e4i)T + (1.54e8 + 2.06e9i)T^{2} \) |
| 79 | \( 1 + (2.29e3 + 3.96e3i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (-4.10e3 + 1.79e4i)T + (-3.54e9 - 1.70e9i)T^{2} \) |
| 89 | \( 1 + (-3.72e4 + 1.14e4i)T + (4.61e9 - 3.14e9i)T^{2} \) |
| 97 | \( 1 - 9.12e4T + 8.58e9T^{2} \) |
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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
−14.42718433602873586672577012322, −13.34563339256262008223656799936, −12.59310147332417220324380437892, −10.17906550193479756083374676938, −9.244905726364889323762203956380, −8.141970370690971625257137536294, −7.52292313305749932742490184996, −5.48751637567315003611777297920, −3.21469701949064386777347428184, −0.72830259998536590222294495474,
2.31040643709955609585325634309, 2.94518783657734954351141087394, 5.86407185589058313087602307593, 7.76620808215030551601804617651, 9.072698234321940221918773803180, 9.886616579207489486604634236792, 10.82757297981514967671865544276, 12.26036730560795468310739361112, 13.86485470128905540842118886697, 14.75909730344354802096034459187
