L(s) = 1 | + (−0.5 − 0.866i)3-s + (−0.5 + 0.866i)5-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)11-s − 13-s + 15-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s + 27-s − 29-s + (0.5 + 0.866i)31-s + (−0.5 + 0.866i)33-s + (−0.5 + 0.866i)37-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)3-s + (−0.5 + 0.866i)5-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)11-s − 13-s + 15-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s + 27-s − 29-s + (0.5 + 0.866i)31-s + (−0.5 + 0.866i)33-s + (−0.5 + 0.866i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2957566945 - 0.3151204055i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2957566945 - 0.3151204055i\) |
\(L(1)\) |
\(\approx\) |
\(0.6411015239 - 0.04068442112i\) |
\(L(1)\) |
\(\approx\) |
\(0.6411015239 - 0.04068442112i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 \) |
good | 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + (0.5 + 0.866i)T \) |
| 19 | \( 1 + (-0.5 + 0.866i)T \) |
| 23 | \( 1 + (0.5 - 0.866i)T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 + (0.5 + 0.866i)T \) |
| 37 | \( 1 + (-0.5 + 0.866i)T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + (-0.5 + 0.866i)T \) |
| 53 | \( 1 + (0.5 + 0.866i)T \) |
| 59 | \( 1 + (0.5 + 0.866i)T \) |
| 61 | \( 1 + (-0.5 + 0.866i)T \) |
| 67 | \( 1 + (-0.5 - 0.866i)T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 + (-0.5 - 0.866i)T \) |
| 79 | \( 1 + (-0.5 + 0.866i)T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 + (0.5 - 0.866i)T \) |
| 97 | \( 1 - T \) |
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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
−21.2378803550600614259586766825, −20.586046343272668295690505433240, −19.98013173727123237627034461426, −19.146780877004034477307640105738, −17.99632143939852950039777727623, −17.212195391842366011686971263076, −16.74629276070275528479727840640, −15.81878858496020994474584509503, −15.2772156194425507538973378877, −14.61447996983324141242982858779, −13.29917010688602519788003964042, −12.58839089129728695903422249631, −11.74459594516319000900914770760, −11.2360319963660772875526268746, −9.990137578338147606294738636849, −9.56880060327708124635930813481, −8.71752651358757708980659230054, −7.6220784593839827473185098300, −6.91078393364906877105845464348, −5.426617723629440067916312129108, −5.04511443979751149369128110421, −4.29059866006030020492369772207, −3.30518018000274349200188693888, −2.08827732599362342362549967604, −0.568977154952572374723702001654,
0.16115298690987952807583905184, 1.47409039776532746241019399023, 2.58542823533733647044930968032, 3.3442559255956144946057620261, 4.612319348499775218121723871931, 5.68217199951886491922703166218, 6.380834503002554474690112781622, 7.20912546567363025978527101257, 7.94982202336672884603302749799, 8.5947062501239019713802947923, 10.214114153431525976356794004784, 10.61769575695488941648260695221, 11.52709845556922364999088945917, 12.24763649023715308384372663687, 12.92295486673869570411202160572, 13.924828269043851873411266202262, 14.60992322494888029202186814913, 15.35153954465327467817012144958, 16.589630751208550348691659904447, 16.917874384284525600447638284554, 18.02372096135271137942459758004, 18.68987598548352095475635490316, 19.16879640979247305734585329332, 19.78135643807554120063718660918, 21.03207664686785185256195186320