L(s) = 1 | + (−0.200 + 0.979i)2-s + (−0.632 − 0.774i)3-s + (−0.919 − 0.391i)4-s + (0.278 + 0.960i)5-s + (0.885 − 0.464i)6-s + (0.568 − 0.822i)8-s + (−0.200 + 0.979i)9-s + (−0.996 + 0.0804i)10-s + (−0.200 − 0.979i)11-s + (0.278 + 0.960i)12-s + (0.568 − 0.822i)15-s + (0.692 + 0.721i)16-s + (0.428 + 0.903i)17-s + (−0.919 − 0.391i)18-s + (−0.5 − 0.866i)19-s + (0.120 − 0.992i)20-s + ⋯ |
L(s) = 1 | + (−0.200 + 0.979i)2-s + (−0.632 − 0.774i)3-s + (−0.919 − 0.391i)4-s + (0.278 + 0.960i)5-s + (0.885 − 0.464i)6-s + (0.568 − 0.822i)8-s + (−0.200 + 0.979i)9-s + (−0.996 + 0.0804i)10-s + (−0.200 − 0.979i)11-s + (0.278 + 0.960i)12-s + (0.568 − 0.822i)15-s + (0.692 + 0.721i)16-s + (0.428 + 0.903i)17-s + (−0.919 − 0.391i)18-s + (−0.5 − 0.866i)19-s + (0.120 − 0.992i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.533 - 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.533 - 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.03558809049 - 0.06450344084i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.03558809049 - 0.06450344084i\) |
\(L(1)\) |
\(\approx\) |
\(0.5673770492 + 0.1773145399i\) |
\(L(1)\) |
\(\approx\) |
\(0.5673770492 + 0.1773145399i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.200 + 0.979i)T \) |
| 3 | \( 1 + (-0.632 - 0.774i)T \) |
| 5 | \( 1 + (0.278 + 0.960i)T \) |
| 11 | \( 1 + (-0.200 - 0.979i)T \) |
| 17 | \( 1 + (0.428 + 0.903i)T \) |
| 19 | \( 1 + (-0.5 - 0.866i)T \) |
| 23 | \( 1 + (-0.5 - 0.866i)T \) |
| 29 | \( 1 + (-0.748 - 0.663i)T \) |
| 31 | \( 1 + (-0.0402 + 0.999i)T \) |
| 37 | \( 1 + (-0.845 - 0.534i)T \) |
| 41 | \( 1 + (-0.354 + 0.935i)T \) |
| 43 | \( 1 + (0.885 + 0.464i)T \) |
| 47 | \( 1 + (-0.919 + 0.391i)T \) |
| 53 | \( 1 + (0.428 + 0.903i)T \) |
| 59 | \( 1 + (0.692 - 0.721i)T \) |
| 61 | \( 1 + (0.428 - 0.903i)T \) |
| 67 | \( 1 + (0.799 + 0.600i)T \) |
| 71 | \( 1 + (-0.354 + 0.935i)T \) |
| 73 | \( 1 + (-0.200 - 0.979i)T \) |
| 79 | \( 1 + (-0.919 + 0.391i)T \) |
| 83 | \( 1 + (-0.354 - 0.935i)T \) |
| 89 | \( 1 + (-0.5 - 0.866i)T \) |
| 97 | \( 1 + (-0.970 + 0.239i)T \) |
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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
−21.20277941367315260356272951140, −20.70571570928926607888636253129, −20.34978373269217216526418679425, −19.32183724344853861121964368200, −18.26535117943250028527973829967, −17.66944830439434074306225535559, −16.927509293771175534923591670787, −16.37595473470146945050022308897, −15.398124220862191967086606507133, −14.42515970929609016439517919072, −13.46407523540987072051822953917, −12.5819473978334830645377408544, −12.04641491469454659541153619360, −11.36495235458828601687458460418, −10.24321000372855596150622208253, −9.82357988271660639647930015651, −9.14746592483482030299734956725, −8.2660765207156695034398594445, −7.175256754091159058684075612227, −5.60952066815252899107463397789, −5.20275955204313323021770993574, −4.245146773270352283383083618844, −3.625449576968061176164502872461, −2.2352712001912682074406338368, −1.29144546396423311592373975581,
0.037545141076371622193470245949, 1.41458835558560399537408871643, 2.62739529441153837500626601306, 3.85420642598469890413742309820, 5.082200349273735104076007197188, 5.96458269122575343771668663354, 6.3927459413864192604933758018, 7.17924101211361591193870166805, 8.01926953547577319373886448583, 8.71099270953233859964757661239, 9.98914338228099585802925276571, 10.70428320213719629302927433141, 11.35739804421936896132974659625, 12.64301178652903527885295569092, 13.267052837694351258157124212944, 14.11550751204916234267222498642, 14.62606350102481317460898007387, 15.691404231240011549969418932, 16.40596411820755625725169565664, 17.302621090835189487200450951981, 17.70552500472494611314497715936, 18.660811528340989762594257370852, 18.97146979314310886828356416826, 19.72198033127918238396228166249, 21.420911019077879469083082555749