L(s) = 1 | + (0.980 + 0.195i)5-s + (−0.382 + 0.923i)7-s + (−0.555 − 0.831i)11-s + (0.980 − 0.195i)13-s + (−0.707 + 0.707i)17-s + (0.195 + 0.980i)19-s + (0.923 − 0.382i)23-s + (0.923 + 0.382i)25-s + (0.555 − 0.831i)29-s + i·31-s + (−0.555 + 0.831i)35-s + (−0.195 + 0.980i)37-s + (−0.923 + 0.382i)41-s + (−0.831 + 0.555i)43-s + (0.707 − 0.707i)47-s + ⋯ |
L(s) = 1 | + (0.980 + 0.195i)5-s + (−0.382 + 0.923i)7-s + (−0.555 − 0.831i)11-s + (0.980 − 0.195i)13-s + (−0.707 + 0.707i)17-s + (0.195 + 0.980i)19-s + (0.923 − 0.382i)23-s + (0.923 + 0.382i)25-s + (0.555 − 0.831i)29-s + i·31-s + (−0.555 + 0.831i)35-s + (−0.195 + 0.980i)37-s + (−0.923 + 0.382i)41-s + (−0.831 + 0.555i)43-s + (0.707 − 0.707i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.146 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.146 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.555733396 + 1.341984856i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.555733396 + 1.341984856i\) |
\(L(1)\) |
\(\approx\) |
\(1.194337800 + 0.2870755416i\) |
\(L(1)\) |
\(\approx\) |
\(1.194337800 + 0.2870755416i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (0.980 + 0.195i)T \) |
| 7 | \( 1 + (-0.382 + 0.923i)T \) |
| 11 | \( 1 + (-0.555 - 0.831i)T \) |
| 13 | \( 1 + (0.980 - 0.195i)T \) |
| 17 | \( 1 + (-0.707 + 0.707i)T \) |
| 19 | \( 1 + (0.195 + 0.980i)T \) |
| 23 | \( 1 + (0.923 - 0.382i)T \) |
| 29 | \( 1 + (0.555 - 0.831i)T \) |
| 31 | \( 1 + iT \) |
| 37 | \( 1 + (-0.195 + 0.980i)T \) |
| 41 | \( 1 + (-0.923 + 0.382i)T \) |
| 43 | \( 1 + (-0.831 + 0.555i)T \) |
| 47 | \( 1 + (0.707 - 0.707i)T \) |
| 53 | \( 1 + (0.555 + 0.831i)T \) |
| 59 | \( 1 + (-0.980 - 0.195i)T \) |
| 61 | \( 1 + (0.831 + 0.555i)T \) |
| 67 | \( 1 + (0.831 + 0.555i)T \) |
| 71 | \( 1 + (-0.382 + 0.923i)T \) |
| 73 | \( 1 + (-0.382 - 0.923i)T \) |
| 79 | \( 1 + (-0.707 - 0.707i)T \) |
| 83 | \( 1 + (0.195 + 0.980i)T \) |
| 89 | \( 1 + (0.923 + 0.382i)T \) |
| 97 | \( 1 + iT \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−24.02748939524298483238410971999, −23.23355544429883301222181305995, −22.41536529115604147582586991275, −21.41083592669187495018837143157, −20.5260551977862999839806659570, −20.03333432376309051747536803212, −18.696346159520533139583261443273, −17.821747932992794019721996544531, −17.1877028080013269823549938962, −16.17464256576237193447539217845, −15.33492663537324483827941108959, −14.00747129199609713831683121473, −13.39091367645610683382815694019, −12.76898416672934736225236797874, −11.27601718600615809464697545703, −10.463030608392245618759671536415, −9.54750960706945988571496158788, −8.75634201050320288548776395553, −7.25186874541914495541705400476, −6.642254180911670647110170653902, −5.349352197190586229555907833382, −4.44560509123928642472004848993, −3.07941434279198471556445528316, −1.866852651304794430937870443, −0.587050115055144969936025366986,
1.28617949791241060677477589817, 2.533426447044540681805691248784, 3.43760516781283782081072162646, 5.10448264391183010723330884216, 5.98978904708914877997646953960, 6.58362800914599046691291385907, 8.31903144897095553617932335682, 8.84884891947978730096512329126, 10.076358769460320535093109866309, 10.780444224016876736465492739695, 11.94495939080973346631809077110, 13.10123440204049838691905179435, 13.57315301322371276874694156990, 14.76287860776523322108688894239, 15.64214261760491281306737809320, 16.53212721166821409411258808646, 17.541254969083832781167054271561, 18.54860428668420659379538261758, 18.86146283282611136852168568761, 20.26848024236678854324237461381, 21.28321718965976592805568767985, 21.68633846533057214262681221622, 22.67639946775646900688002873207, 23.57763877334064111045471192240, 24.85805627779640183507081631930