L(s) = 1 | + (−0.173 − 0.984i)2-s + (−0.835 − 0.549i)3-s + (−0.939 + 0.342i)4-s + (−0.396 + 0.918i)5-s + (−0.396 + 0.918i)6-s + (−0.993 + 0.116i)7-s + (0.5 + 0.866i)8-s + (0.396 + 0.918i)9-s + (0.973 + 0.230i)10-s + (0.973 − 0.230i)11-s + (0.973 + 0.230i)12-s + (−0.597 − 0.802i)13-s + (0.286 + 0.957i)14-s + (0.835 − 0.549i)15-s + (0.766 − 0.642i)16-s + (−0.173 − 0.984i)17-s + ⋯ |
L(s) = 1 | + (−0.173 − 0.984i)2-s + (−0.835 − 0.549i)3-s + (−0.939 + 0.342i)4-s + (−0.396 + 0.918i)5-s + (−0.396 + 0.918i)6-s + (−0.993 + 0.116i)7-s + (0.5 + 0.866i)8-s + (0.396 + 0.918i)9-s + (0.973 + 0.230i)10-s + (0.973 − 0.230i)11-s + (0.973 + 0.230i)12-s + (−0.597 − 0.802i)13-s + (0.286 + 0.957i)14-s + (0.835 − 0.549i)15-s + (0.766 − 0.642i)16-s + (−0.173 − 0.984i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.879 - 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.879 - 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1547414098 - 0.6116638192i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1547414098 - 0.6116638192i\) |
\(L(1)\) |
\(\approx\) |
\(0.4921690065 - 0.3218553390i\) |
\(L(1)\) |
\(\approx\) |
\(0.4921690065 - 0.3218553390i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
| 109 | \( 1 \) |
good | 2 | \( 1 + (-0.173 - 0.984i)T \) |
| 3 | \( 1 + (-0.835 - 0.549i)T \) |
| 5 | \( 1 + (-0.396 + 0.918i)T \) |
| 7 | \( 1 + (-0.993 + 0.116i)T \) |
| 11 | \( 1 + (0.973 - 0.230i)T \) |
| 13 | \( 1 + (-0.597 - 0.802i)T \) |
| 17 | \( 1 + (-0.173 - 0.984i)T \) |
| 19 | \( 1 + (0.5 - 0.866i)T \) |
| 23 | \( 1 + (0.939 + 0.342i)T \) |
| 29 | \( 1 + (-0.973 - 0.230i)T \) |
| 31 | \( 1 + (0.835 - 0.549i)T \) |
| 41 | \( 1 + (0.173 + 0.984i)T \) |
| 43 | \( 1 + (0.939 - 0.342i)T \) |
| 47 | \( 1 + (-0.993 - 0.116i)T \) |
| 53 | \( 1 + (0.973 + 0.230i)T \) |
| 59 | \( 1 + (0.835 - 0.549i)T \) |
| 61 | \( 1 + (0.286 + 0.957i)T \) |
| 67 | \( 1 + (-0.286 + 0.957i)T \) |
| 71 | \( 1 + (0.173 - 0.984i)T \) |
| 73 | \( 1 + (-0.686 - 0.727i)T \) |
| 79 | \( 1 + (0.686 + 0.727i)T \) |
| 83 | \( 1 + (-0.993 + 0.116i)T \) |
| 89 | \( 1 + (-0.893 + 0.448i)T \) |
| 97 | \( 1 + (-0.893 + 0.448i)T \) |
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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
−18.76800655993053364595033304053, −17.66965805570789637339022727836, −17.02683678514520483883355933494, −16.762919477155453203474327791342, −16.19361555881941638094983208163, −15.60491431139653446998695881217, −14.86108951945086865929493356253, −14.252734426691175101268140997012, −13.183748761545463675954564291791, −12.55931763104181460234995157794, −12.11434607374187457398876636579, −11.18897829790840777923267453205, −10.177666474295294792356091594735, −9.624929379408753211725897270630, −9.08478589225031075467485767253, −8.475201679228965727566328059391, −7.2856342308794495971408949053, −6.829994048355133480705757407255, −6.04775951052120498241106656554, −5.456274646438190899465473697821, −4.575190174074584280789590369260, −4.05795874754128717751362185858, −3.47332825820420320032612310164, −1.5998222017546503782227201814, −0.77766034904275002358580088332,
0.36107213471909514476946983595, 1.05487829601673403164265101287, 2.41618192436387432712067544444, 2.8473215952996725123540012757, 3.65645414905360907534584714737, 4.56250845927511773072663551672, 5.43210028780115257191039867542, 6.21714061820428007435107339541, 7.1575181965724164487599396991, 7.39767655360121881493830867546, 8.53756369038541366079096475157, 9.54638740686370610319180778418, 9.89310699699100660230785301524, 10.820601747270980172832629096101, 11.44064856004527077600308325964, 11.7706456560870892208914363813, 12.55605668555984858580237804176, 13.29620124391175373364143411652, 13.706672388395992535142189707968, 14.72226832533750751703852502720, 15.47360511577309322417013064197, 16.38236641686020503965255981134, 17.013365912420999273812877638772, 17.76976311451217337066007194069, 18.203239419106028969062008663138