| L(s) = 1 | + (−3.23e3 + 5.61e3i)5-s + (−4.40e4 + 6.34e3i)7-s + (−1.53e5 − 2.66e5i)11-s + 5.84e5·13-s + (−4.72e6 − 8.17e6i)17-s + (8.44e6 − 1.46e7i)19-s + (−1.59e7 + 2.76e7i)23-s + (3.42e6 + 5.93e6i)25-s + 1.08e8·29-s + (1.25e8 + 2.17e8i)31-s + (1.06e8 − 2.67e8i)35-s + (8.03e7 − 1.39e8i)37-s + 5.43e8·41-s + 2.35e8·43-s + (−4.31e8 + 7.46e8i)47-s + ⋯ |
| L(s) = 1 | + (−0.463 + 0.802i)5-s + (−0.989 + 0.142i)7-s + (−0.288 − 0.498i)11-s + 0.436·13-s + (−0.806 − 1.39i)17-s + (0.782 − 1.35i)19-s + (−0.516 + 0.894i)23-s + (0.0701 + 0.121i)25-s + 0.977·29-s + (0.787 + 1.36i)31-s + (0.344 − 0.860i)35-s + (0.190 − 0.330i)37-s + 0.732·41-s + 0.244·43-s + (−0.274 + 0.474i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.950 - 0.311i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.950 - 0.311i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.3946368750\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3946368750\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (4.40e4 - 6.34e3i)T \) |
| good | 5 | \( 1 + (3.23e3 - 5.61e3i)T + (-2.44e7 - 4.22e7i)T^{2} \) |
| 11 | \( 1 + (1.53e5 + 2.66e5i)T + (-1.42e11 + 2.47e11i)T^{2} \) |
| 13 | \( 1 - 5.84e5T + 1.79e12T^{2} \) |
| 17 | \( 1 + (4.72e6 + 8.17e6i)T + (-1.71e13 + 2.96e13i)T^{2} \) |
| 19 | \( 1 + (-8.44e6 + 1.46e7i)T + (-5.82e13 - 1.00e14i)T^{2} \) |
| 23 | \( 1 + (1.59e7 - 2.76e7i)T + (-4.76e14 - 8.25e14i)T^{2} \) |
| 29 | \( 1 - 1.08e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + (-1.25e8 - 2.17e8i)T + (-1.27e16 + 2.20e16i)T^{2} \) |
| 37 | \( 1 + (-8.03e7 + 1.39e8i)T + (-8.89e16 - 1.54e17i)T^{2} \) |
| 41 | \( 1 - 5.43e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 2.35e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + (4.31e8 - 7.46e8i)T + (-1.23e18 - 2.14e18i)T^{2} \) |
| 53 | \( 1 + (-1.83e9 - 3.18e9i)T + (-4.63e18 + 8.02e18i)T^{2} \) |
| 59 | \( 1 + (-1.91e9 - 3.31e9i)T + (-1.50e19 + 2.61e19i)T^{2} \) |
| 61 | \( 1 + (2.30e9 - 3.98e9i)T + (-2.17e19 - 3.76e19i)T^{2} \) |
| 67 | \( 1 + (-9.21e8 - 1.59e9i)T + (-6.10e19 + 1.05e20i)T^{2} \) |
| 71 | \( 1 + 1.51e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + (1.51e10 + 2.61e10i)T + (-1.56e20 + 2.71e20i)T^{2} \) |
| 79 | \( 1 + (-1.74e10 + 3.02e10i)T + (-3.73e20 - 6.47e20i)T^{2} \) |
| 83 | \( 1 + 2.03e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + (4.42e10 - 7.65e10i)T + (-1.38e21 - 2.40e21i)T^{2} \) |
| 97 | \( 1 + 9.58e10T + 7.15e21T^{2} \) |
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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
−10.63850381024914722408742554104, −9.531067028950149211903411976299, −8.762516454380714276124159609249, −7.37167821307391343820520431541, −6.81300156861674031574920304427, −5.73008792521473435339716323138, −4.47651749098710410074608109490, −3.07068181388528635600842842413, −2.80998237716955647841756826733, −0.917944685101398425773061501388,
0.093478255536231775608334972651, 1.11322957608700849650511990241, 2.40893457963332002491962878965, 3.77713839551943497301029018147, 4.43389550595738850099778287300, 5.83242023125788518440408722622, 6.62282781620541753140359359238, 7.977111881479739970812490932482, 8.561213115981380734610732525576, 9.791870520889038959272748730976
