L(s) = 1 | + (0.445 − 0.895i)2-s + (−0.982 + 0.183i)3-s + (−0.602 − 0.798i)4-s + (−0.982 + 0.183i)5-s + (−0.273 + 0.961i)6-s + (−0.273 + 0.961i)7-s + (−0.982 + 0.183i)8-s + (0.932 − 0.361i)9-s + (−0.273 + 0.961i)10-s + (0.932 + 0.361i)11-s + (0.739 + 0.673i)12-s + (−0.602 + 0.798i)13-s + (0.739 + 0.673i)14-s + (0.932 − 0.361i)15-s + (−0.273 + 0.961i)16-s + (0.0922 − 0.995i)17-s + ⋯ |
L(s) = 1 | + (0.445 − 0.895i)2-s + (−0.982 + 0.183i)3-s + (−0.602 − 0.798i)4-s + (−0.982 + 0.183i)5-s + (−0.273 + 0.961i)6-s + (−0.273 + 0.961i)7-s + (−0.982 + 0.183i)8-s + (0.932 − 0.361i)9-s + (−0.273 + 0.961i)10-s + (0.932 + 0.361i)11-s + (0.739 + 0.673i)12-s + (−0.602 + 0.798i)13-s + (0.739 + 0.673i)14-s + (0.932 − 0.361i)15-s + (−0.273 + 0.961i)16-s + (0.0922 − 0.995i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 919 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.962 + 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 919 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.962 + 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.01799065741 - 0.1298581036i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.01799065741 - 0.1298581036i\) |
\(L(1)\) |
\(\approx\) |
\(0.5887341663 - 0.1981191723i\) |
\(L(1)\) |
\(\approx\) |
\(0.5887341663 - 0.1981191723i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 919 | \( 1 \) |
good | 2 | \( 1 + (0.445 - 0.895i)T \) |
| 3 | \( 1 + (-0.982 + 0.183i)T \) |
| 5 | \( 1 + (-0.982 + 0.183i)T \) |
| 7 | \( 1 + (-0.273 + 0.961i)T \) |
| 11 | \( 1 + (0.932 + 0.361i)T \) |
| 13 | \( 1 + (-0.602 + 0.798i)T \) |
| 17 | \( 1 + (0.0922 - 0.995i)T \) |
| 19 | \( 1 + (-0.982 - 0.183i)T \) |
| 23 | \( 1 + (0.932 + 0.361i)T \) |
| 29 | \( 1 + (0.445 - 0.895i)T \) |
| 31 | \( 1 + (-0.273 + 0.961i)T \) |
| 37 | \( 1 + (-0.602 + 0.798i)T \) |
| 41 | \( 1 + (0.739 - 0.673i)T \) |
| 43 | \( 1 + (-0.982 - 0.183i)T \) |
| 47 | \( 1 + (-0.602 - 0.798i)T \) |
| 53 | \( 1 + T \) |
| 59 | \( 1 + (-0.602 - 0.798i)T \) |
| 61 | \( 1 + (-0.273 - 0.961i)T \) |
| 67 | \( 1 + (-0.850 + 0.526i)T \) |
| 71 | \( 1 + (-0.273 + 0.961i)T \) |
| 73 | \( 1 + (-0.850 + 0.526i)T \) |
| 79 | \( 1 + (-0.273 - 0.961i)T \) |
| 83 | \( 1 + (0.0922 + 0.995i)T \) |
| 89 | \( 1 + (-0.982 - 0.183i)T \) |
| 97 | \( 1 + (0.932 - 0.361i)T \) |
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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
−22.825577088185378074842163790408, −21.85025994943206494240702887053, −20.991616390672055995733370204320, −19.71337015860090640483648052309, −19.22767415784498168241868973125, −18.07263869764711653292748376918, −17.1584335517864566776478140975, −16.74942656730501251046343170692, −16.217866237768174894591335701, −15.0810943227742087618283775398, −14.64758035724549380286599026193, −13.30448525953917132299008897397, −12.71739225868510380581094080536, −12.10062959403223822634529221504, −11.072024859892834544333834891989, −10.34559367202745730363183944888, −9.02472349031577235071004478266, −8.02807725255827509043196390648, −7.29819449040639319850144152018, −6.61080973210137416706943933906, −5.81007131926972349237620227193, −4.648693077832644116245038692842, −4.14077106903952610665604614464, −3.22893798058884013183336750464, −1.10994806623897185849228849972,
0.068270425630645068344951090811, 1.51621176577156589965808870940, 2.71507189645326446455983189635, 3.78032306600796991394034377738, 4.61636812975368862317266508177, 5.23282795607628358825856435573, 6.486340195749313773979617551730, 7.009767916613570209923740827881, 8.68163275408108170935138914724, 9.38544664222427918819782138685, 10.25120551513836236803451718941, 11.29402590638047107029921831175, 11.82831947146922101200819719909, 12.17627183420181043687066263694, 13.05902470556543910133743427636, 14.33897347612433874842760723676, 15.11259310201361190999116686188, 15.69297593550538670010726380230, 16.72860905425510724260112099939, 17.611820445775760010150016507, 18.60010655510634369337427866062, 19.11517350890786205952735385713, 19.73818359074371641080546969105, 20.86781284112145674748063345230, 21.67344378510751685600454120548