L(s) = 1 | + (0.453 − 0.891i)2-s + (−0.891 + 0.453i)3-s + (−0.587 − 0.809i)4-s + i·6-s + (−0.0784 + 0.996i)7-s + (−0.987 + 0.156i)8-s + (0.587 − 0.809i)9-s + (−0.996 − 0.0784i)11-s + (0.891 + 0.453i)12-s + (0.649 + 0.760i)13-s + (0.852 + 0.522i)14-s + (−0.309 + 0.951i)16-s + (0.0784 + 0.996i)17-s + (−0.453 − 0.891i)18-s + (0.522 + 0.852i)19-s + ⋯ |
L(s) = 1 | + (0.453 − 0.891i)2-s + (−0.891 + 0.453i)3-s + (−0.587 − 0.809i)4-s + i·6-s + (−0.0784 + 0.996i)7-s + (−0.987 + 0.156i)8-s + (0.587 − 0.809i)9-s + (−0.996 − 0.0784i)11-s + (0.891 + 0.453i)12-s + (0.649 + 0.760i)13-s + (0.852 + 0.522i)14-s + (−0.309 + 0.951i)16-s + (0.0784 + 0.996i)17-s + (−0.453 − 0.891i)18-s + (0.522 + 0.852i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.935 + 0.353i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.935 + 0.353i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.140970405 + 0.2086405535i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.140970405 + 0.2086405535i\) |
\(L(1)\) |
\(\approx\) |
\(0.8635037916 - 0.1600034912i\) |
\(L(1)\) |
\(\approx\) |
\(0.8635037916 - 0.1600034912i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 241 | \( 1 \) |
good | 2 | \( 1 + (0.453 - 0.891i)T \) |
| 3 | \( 1 + (-0.891 + 0.453i)T \) |
| 7 | \( 1 + (-0.0784 + 0.996i)T \) |
| 11 | \( 1 + (-0.996 - 0.0784i)T \) |
| 13 | \( 1 + (0.649 + 0.760i)T \) |
| 17 | \( 1 + (0.0784 + 0.996i)T \) |
| 19 | \( 1 + (0.522 + 0.852i)T \) |
| 23 | \( 1 + (-0.522 - 0.852i)T \) |
| 29 | \( 1 + (0.453 - 0.891i)T \) |
| 31 | \( 1 + (-0.996 + 0.0784i)T \) |
| 37 | \( 1 + (0.972 - 0.233i)T \) |
| 41 | \( 1 + (0.707 - 0.707i)T \) |
| 43 | \( 1 + (-0.996 + 0.0784i)T \) |
| 47 | \( 1 + (0.891 + 0.453i)T \) |
| 53 | \( 1 + (0.453 - 0.891i)T \) |
| 59 | \( 1 + (0.156 - 0.987i)T \) |
| 61 | \( 1 + (-0.156 - 0.987i)T \) |
| 67 | \( 1 + (0.891 - 0.453i)T \) |
| 71 | \( 1 + (-0.760 + 0.649i)T \) |
| 73 | \( 1 + (0.649 - 0.760i)T \) |
| 79 | \( 1 + (0.707 - 0.707i)T \) |
| 83 | \( 1 + (0.309 + 0.951i)T \) |
| 89 | \( 1 + (-0.760 + 0.649i)T \) |
| 97 | \( 1 + (0.309 + 0.951i)T \) |
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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
−17.74281587122922461732471852812, −16.81040752808587240867362351580, −16.43247086690086269209311499445, −15.76377943416479849844126153977, −15.34151660989004305083199406010, −14.218552213914332338723617589097, −13.61064467569584518645723994220, −13.17085569412760589093596323370, −12.7316415754518718343789172115, −11.75141922984629353806981287881, −11.21638594345971168067229962186, −10.425752696935964873014781515988, −9.78710668693080436977917544930, −8.81300907165278226551036249473, −7.87128661394872099203913828312, −7.39798206950910220145551916444, −7.04245241295748522201637693236, −6.091759794138493933632629126298, −5.49506382918233564605040475497, −4.94545614633573688896730279619, −4.24834518060231871693744051123, −3.316629217673171306111644796986, −2.590617888557830520759515644722, −1.17692402202206461695664128759, −0.437909008600572300107115362995,
0.716700744183629041089382350826, 1.82270447249353567260269046409, 2.36124267979117973163884914283, 3.490576365827943347463347075467, 3.94813100100457202447240612153, 4.83561044902507338002439049433, 5.43127955657606552333644618738, 6.09135122446278986701633073882, 6.42867151210442692281771627222, 7.85653630411002426774247899959, 8.59923965250510672631018972903, 9.36056580977767244477135736493, 9.972401795421436840991729259184, 10.59870486587003779753936416923, 11.19883793449709233034238105131, 11.7679918302325603839668771917, 12.53527138597851172933558468544, 12.73882934252246161177123174320, 13.701376296743501421852630580836, 14.51650028639451606718621209190, 15.08634692192804466602304662877, 15.85257611643654907738333675533, 16.22828143641653027783949311033, 17.16110879464917908885940150293, 18.04251259360549065665110179795