L(s) = 1 | + (−0.173 + 0.984i)2-s + (0.993 − 0.116i)3-s + (−0.939 − 0.342i)4-s + (0.957 − 0.286i)5-s + (−0.0581 + 0.998i)6-s + (−0.686 + 0.727i)7-s + (0.5 − 0.866i)8-s + (0.973 − 0.230i)9-s + (0.116 + 0.993i)10-s + (−0.116 + 0.993i)11-s + (−0.973 − 0.230i)12-s + (0.686 − 0.727i)13-s + (−0.597 − 0.802i)14-s + (0.918 − 0.396i)15-s + (0.766 + 0.642i)16-s + (0.766 + 0.642i)17-s + ⋯ |
L(s) = 1 | + (−0.173 + 0.984i)2-s + (0.993 − 0.116i)3-s + (−0.939 − 0.342i)4-s + (0.957 − 0.286i)5-s + (−0.0581 + 0.998i)6-s + (−0.686 + 0.727i)7-s + (0.5 − 0.866i)8-s + (0.973 − 0.230i)9-s + (0.116 + 0.993i)10-s + (−0.116 + 0.993i)11-s + (−0.973 − 0.230i)12-s + (0.686 − 0.727i)13-s + (−0.597 − 0.802i)14-s + (0.918 − 0.396i)15-s + (0.766 + 0.642i)16-s + (0.766 + 0.642i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.470 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.470 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.447905012 + 1.469301660i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.447905012 + 1.469301660i\) |
\(L(1)\) |
\(\approx\) |
\(1.433672418 + 0.6403797249i\) |
\(L(1)\) |
\(\approx\) |
\(1.433672418 + 0.6403797249i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
| 109 | \( 1 \) |
good | 2 | \( 1 + (-0.173 + 0.984i)T \) |
| 3 | \( 1 + (0.993 - 0.116i)T \) |
| 5 | \( 1 + (0.957 - 0.286i)T \) |
| 7 | \( 1 + (-0.686 + 0.727i)T \) |
| 11 | \( 1 + (-0.116 + 0.993i)T \) |
| 13 | \( 1 + (0.686 - 0.727i)T \) |
| 17 | \( 1 + (0.766 + 0.642i)T \) |
| 19 | \( 1 + (0.939 - 0.342i)T \) |
| 23 | \( 1 + (-0.939 - 0.342i)T \) |
| 29 | \( 1 + (0.918 + 0.396i)T \) |
| 31 | \( 1 + (0.727 - 0.686i)T \) |
| 41 | \( 1 + (0.866 - 0.5i)T \) |
| 43 | \( 1 + (-0.984 - 0.173i)T \) |
| 47 | \( 1 + (0.549 + 0.835i)T \) |
| 53 | \( 1 + (-0.957 + 0.286i)T \) |
| 59 | \( 1 + (-0.396 - 0.918i)T \) |
| 61 | \( 1 + (0.549 - 0.835i)T \) |
| 67 | \( 1 + (-0.727 + 0.686i)T \) |
| 71 | \( 1 + (-0.173 - 0.984i)T \) |
| 73 | \( 1 + (0.993 + 0.116i)T \) |
| 79 | \( 1 + (-0.973 + 0.230i)T \) |
| 83 | \( 1 + (-0.597 - 0.802i)T \) |
| 89 | \( 1 + (0.998 + 0.0581i)T \) |
| 97 | \( 1 + (-0.230 + 0.973i)T \) |
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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
−18.28683794611066450630977400764, −18.20989987729080230429181321479, −16.99985279292127950301095290569, −16.3416104410911181217916097338, −15.8331213944941660571390779657, −14.42876283303750559113963057473, −13.91645897849575532680417266595, −13.729476377144840586105003003760, −13.141192935589520844647270743905, −12.18771529533507707207794653077, −11.408498004048378143879768606948, −10.488534692167628234370581210740, −9.995794066425735139018639635713, −9.56653894868270924024694234191, −8.82634054909140283854050986533, −8.12521664165742746814340509165, −7.32314243277635109668798209538, −6.40610618840367424430110874550, −5.55652234075513670406511850311, −4.536109569823573803993486879109, −3.69301300790917149368050632371, −3.1302601017763027795904090281, −2.62306596135461868560540888410, −1.5029981449803441769033708005, −0.99565347114942213912369496049,
0.94355252834409354173525419633, 1.796507828636842687789872194938, 2.730317998243699529905692959874, 3.51243500694599122730373503023, 4.50570469415430223308598843144, 5.283666221378953216440606254650, 6.08834164231436140448253862810, 6.53478680005022409416832694590, 7.53751686839564443103421827351, 8.15219809521768474698454218788, 8.79425840585925601043596856070, 9.53659431647865691736533015989, 9.89465821066464562654789720976, 10.49119950519221465885845739700, 12.239915568026997027612661776142, 12.66277157082387369027093893984, 13.27723299817757656395443914064, 13.95258555525479915228515952535, 14.492648797125103634394212067115, 15.283897785453956291581890397764, 15.796055803249138230583854197517, 16.30325360849170244793873353890, 17.41887018745192953789860043629, 17.82600676043241120926357424487, 18.517920903901074292950256468313