L(s) = 1 | + (0.932 + 0.361i)2-s + (0.273 + 0.961i)3-s + (0.739 + 0.673i)4-s + (−0.932 + 0.361i)5-s + (−0.0922 + 0.995i)6-s + (−0.602 − 0.798i)7-s + (0.445 + 0.895i)8-s + (−0.850 + 0.526i)9-s − 10-s + (0.739 + 0.673i)11-s + (−0.445 + 0.895i)12-s + (0.602 + 0.798i)13-s + (−0.273 − 0.961i)14-s + (−0.602 − 0.798i)15-s + (0.0922 + 0.995i)16-s + (0.445 − 0.895i)17-s + ⋯ |
L(s) = 1 | + (0.932 + 0.361i)2-s + (0.273 + 0.961i)3-s + (0.739 + 0.673i)4-s + (−0.932 + 0.361i)5-s + (−0.0922 + 0.995i)6-s + (−0.602 − 0.798i)7-s + (0.445 + 0.895i)8-s + (−0.850 + 0.526i)9-s − 10-s + (0.739 + 0.673i)11-s + (−0.445 + 0.895i)12-s + (0.602 + 0.798i)13-s + (−0.273 − 0.961i)14-s + (−0.602 − 0.798i)15-s + (0.0922 + 0.995i)16-s + (0.445 − 0.895i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.300 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.300 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9852567009 + 1.343438318i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9852567009 + 1.343438318i\) |
\(L(1)\) |
\(\approx\) |
\(1.286438721 + 0.9090389139i\) |
\(L(1)\) |
\(\approx\) |
\(1.286438721 + 0.9090389139i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 137 | \( 1 \) |
good | 2 | \( 1 + (0.932 + 0.361i)T \) |
| 3 | \( 1 + (0.273 + 0.961i)T \) |
| 5 | \( 1 + (-0.932 + 0.361i)T \) |
| 7 | \( 1 + (-0.602 - 0.798i)T \) |
| 11 | \( 1 + (0.739 + 0.673i)T \) |
| 13 | \( 1 + (0.602 + 0.798i)T \) |
| 17 | \( 1 + (0.445 - 0.895i)T \) |
| 19 | \( 1 + (-0.982 + 0.183i)T \) |
| 23 | \( 1 + (-0.0922 - 0.995i)T \) |
| 29 | \( 1 + (-0.0922 - 0.995i)T \) |
| 31 | \( 1 + (0.982 + 0.183i)T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 + (0.982 - 0.183i)T \) |
| 47 | \( 1 + (0.850 - 0.526i)T \) |
| 53 | \( 1 + (0.982 - 0.183i)T \) |
| 59 | \( 1 + (-0.850 + 0.526i)T \) |
| 61 | \( 1 + (-0.850 - 0.526i)T \) |
| 67 | \( 1 + (0.602 + 0.798i)T \) |
| 71 | \( 1 + (-0.739 + 0.673i)T \) |
| 73 | \( 1 + (-0.602 + 0.798i)T \) |
| 79 | \( 1 + (0.273 - 0.961i)T \) |
| 83 | \( 1 + (-0.445 - 0.895i)T \) |
| 89 | \( 1 + (-0.932 + 0.361i)T \) |
| 97 | \( 1 + (-0.739 - 0.673i)T \) |
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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
−28.34270669938603727414253719028, −27.617188108427542220097541605733, −25.721576778605101834841081628046, −25.02573662557398448148177840252, −24.008403055524215177933506140699, −23.35221069003219217554017665586, −22.37798824425870859551983429366, −21.226239458432560002596250438881, −19.89765697241221302740054839716, −19.40549637867435313749135815487, −18.59325363472508348861273559232, −16.85929989369427689025439364989, −15.5759326443898454932489604908, −14.8220852990783317156815524199, −13.487417593357753383427990895512, −12.62350677753681202245700217652, −11.968061304516183830156403905550, −10.90544031105277577020784512169, −9.019365277483483821739715567682, −7.93520379964071210880613742715, −6.49325156877105736949739349251, −5.67043986527862519814773272130, −3.83157182308927080108552967221, −2.92411959608529928061459549732, −1.23853079491852993314257825470,
2.763334463025640213508087617715, 4.075986496295229051900514600454, 4.35124679972519331167086297764, 6.28842054219661361049888190832, 7.29519570789059638019308677353, 8.59276937298582033851472697733, 10.119368121934552899461612905059, 11.23142677400897992976606255326, 12.17715005241528054145035872168, 13.69064235588233217948633741921, 14.52632599260166241578976844344, 15.430523990353067522170454889760, 16.342928071655254379114444067041, 17.00475646580919632766999046678, 19.07395873140829405066083956713, 20.11023477513851369325370799391, 20.785641933427883707511869463010, 22.02642290779768538879491573448, 22.995681096913386346859485014094, 23.27678451871411343690582308498, 24.873147040477363822663574196143, 25.93294132525105051286986144997, 26.56275382870275303774178177484, 27.59140178126999107363020154785, 28.81282496732702042590115967259