L(s) = 1 | + (0.236 + 1.39i)2-s + (0.827 + 0.221i)3-s + (−1.88 + 0.659i)4-s + (4.10 − 1.10i)5-s + (−0.113 + 1.20i)6-s + (−2.50 + 0.856i)7-s + (−1.36 − 2.47i)8-s + (−1.96 − 1.13i)9-s + (2.50 + 5.46i)10-s + (−0.318 + 1.18i)11-s + (−1.70 + 0.127i)12-s + (−1.73 + 1.73i)13-s + (−1.78 − 3.28i)14-s + 3.64·15-s + (3.12 − 2.49i)16-s + (−0.931 − 1.61i)17-s + ⋯ |
L(s) = 1 | + (0.167 + 0.985i)2-s + (0.477 + 0.128i)3-s + (−0.944 + 0.329i)4-s + (1.83 − 0.492i)5-s + (−0.0463 + 0.492i)6-s + (−0.946 + 0.323i)7-s + (−0.483 − 0.875i)8-s + (−0.653 − 0.377i)9-s + (0.792 + 1.72i)10-s + (−0.0960 + 0.358i)11-s + (−0.493 + 0.0367i)12-s + (−0.480 + 0.480i)13-s + (−0.477 − 0.878i)14-s + 0.941·15-s + (0.782 − 0.622i)16-s + (−0.226 − 0.391i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.365 - 0.930i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.365 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.07234 + 0.730674i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.07234 + 0.730674i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.236 - 1.39i)T \) |
| 7 | \( 1 + (2.50 - 0.856i)T \) |
good | 3 | \( 1 + (-0.827 - 0.221i)T + (2.59 + 1.5i)T^{2} \) |
| 5 | \( 1 + (-4.10 + 1.10i)T + (4.33 - 2.5i)T^{2} \) |
| 11 | \( 1 + (0.318 - 1.18i)T + (-9.52 - 5.5i)T^{2} \) |
| 13 | \( 1 + (1.73 - 1.73i)T - 13iT^{2} \) |
| 17 | \( 1 + (0.931 + 1.61i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.989 + 3.69i)T + (-16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (-1.23 - 0.711i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.181 - 0.181i)T - 29iT^{2} \) |
| 31 | \( 1 + (-3.23 - 5.59i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.237 + 0.0637i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 - 0.440iT - 41T^{2} \) |
| 43 | \( 1 + (5.54 + 5.54i)T + 43iT^{2} \) |
| 47 | \( 1 + (3.61 - 6.26i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (2.59 - 9.69i)T + (-45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (0.438 - 1.63i)T + (-51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (-2.41 - 9.01i)T + (-52.8 + 30.5i)T^{2} \) |
| 67 | \( 1 + (-9.59 - 2.57i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + 11.2iT - 71T^{2} \) |
| 73 | \( 1 + (8.13 - 4.69i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.52 + 11.3i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (3.06 - 3.06i)T - 83iT^{2} \) |
| 89 | \( 1 + (5.66 + 3.26i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 1.70T + 97T^{2} \) |
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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
−13.83881648241639632068327605590, −13.22389477910623338986127915785, −12.25635735068439559345704059712, −10.05342084323695897180433000781, −9.258470421225472381156409325817, −8.790346963326190625961927317539, −6.86209508217815975867824533758, −5.99767104653003308159553400900, −4.89599327448056712180847794046, −2.77951624861275821114461978880,
2.18620792825539896361286342099, 3.23318629645424331622198228323, 5.39909432672569651078986897397, 6.37672256914359880669981747264, 8.365371306920116968752507478607, 9.626913169972169685853831932776, 10.10963551675717091057975209578, 11.12778441881589038072181549834, 12.80778720741304024736171073852, 13.33604085354823084713835919561