L(s) = 1 | + (−0.616 − 0.787i)2-s + (−0.239 + 0.970i)4-s + (0.995 − 0.0904i)5-s + (0.685 + 0.728i)7-s + (0.911 − 0.410i)8-s + (−0.685 − 0.728i)10-s + (0.0904 − 0.995i)11-s + (0.239 − 0.970i)13-s + (0.150 − 0.988i)14-s + (−0.885 − 0.464i)16-s + (−0.983 − 0.180i)19-s + (−0.150 + 0.988i)20-s + (−0.839 + 0.542i)22-s + (0.382 + 0.923i)23-s + (0.983 − 0.180i)25-s + (−0.911 + 0.410i)26-s + ⋯ |
L(s) = 1 | + (−0.616 − 0.787i)2-s + (−0.239 + 0.970i)4-s + (0.995 − 0.0904i)5-s + (0.685 + 0.728i)7-s + (0.911 − 0.410i)8-s + (−0.685 − 0.728i)10-s + (0.0904 − 0.995i)11-s + (0.239 − 0.970i)13-s + (0.150 − 0.988i)14-s + (−0.885 − 0.464i)16-s + (−0.983 − 0.180i)19-s + (−0.150 + 0.988i)20-s + (−0.839 + 0.542i)22-s + (0.382 + 0.923i)23-s + (0.983 − 0.180i)25-s + (−0.911 + 0.410i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.414 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.414 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.412051231 - 0.9079477012i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.412051231 - 0.9079477012i\) |
\(L(1)\) |
\(\approx\) |
\(0.9680716187 - 0.3512693096i\) |
\(L(1)\) |
\(\approx\) |
\(0.9680716187 - 0.3512693096i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 17 | \( 1 \) |
| 79 | \( 1 \) |
good | 2 | \( 1 + (-0.616 - 0.787i)T \) |
| 5 | \( 1 + (0.995 - 0.0904i)T \) |
| 7 | \( 1 + (0.685 + 0.728i)T \) |
| 11 | \( 1 + (0.0904 - 0.995i)T \) |
| 13 | \( 1 + (0.239 - 0.970i)T \) |
| 19 | \( 1 + (-0.983 - 0.180i)T \) |
| 23 | \( 1 + (0.382 + 0.923i)T \) |
| 29 | \( 1 + (0.437 + 0.899i)T \) |
| 31 | \( 1 + (-0.871 - 0.491i)T \) |
| 37 | \( 1 + (-0.977 + 0.209i)T \) |
| 41 | \( 1 + (0.640 - 0.768i)T \) |
| 43 | \( 1 + (-0.954 - 0.297i)T \) |
| 47 | \( 1 + (-0.822 - 0.568i)T \) |
| 53 | \( 1 + (0.410 + 0.911i)T \) |
| 59 | \( 1 + (0.855 + 0.517i)T \) |
| 61 | \( 1 + (0.839 + 0.542i)T \) |
| 67 | \( 1 + (-0.120 - 0.992i)T \) |
| 71 | \( 1 + (0.728 + 0.685i)T \) |
| 73 | \( 1 + (0.988 + 0.150i)T \) |
| 83 | \( 1 + (0.855 - 0.517i)T \) |
| 89 | \( 1 + (0.935 - 0.354i)T \) |
| 97 | \( 1 + (0.209 + 0.977i)T \) |
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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
−18.28309141548457710718329376335, −17.82281068136683264631482526680, −17.24089616249934573217901845329, −16.72067060432725104975105547807, −16.12352920590465573508067784959, −15.01035805800252736662010459576, −14.55415118627861893305200596982, −14.1170911630405487890630081700, −13.31456974013248313519374303168, −12.65216552620226426566812227808, −11.451197232731722355136499737952, −10.80373510460039358836479124823, −10.11310953204729697169137452919, −9.64206842246796866154816546667, −8.77907159236016542620312482083, −8.228442243898145781367632893299, −7.26539349478892650926038458179, −6.65881223356754728070687927703, −6.2532889109186234546770814203, −5.02992966457009748990666056550, −4.73013235514031477001725009346, −3.80147107603833430327485263275, −2.125957691500568355195925717554, −1.88995609971732592474377584621, −0.8831337361764027530894344784,
0.70967798125184202785756958537, 1.61093532016648606692257025688, 2.237537259105490919650906659038, 3.060514007813593898032056403091, 3.75728958090223192737352841772, 5.02022278112391635230588564569, 5.45954951371033758158350434822, 6.34368820249758900167089085544, 7.30195693364043711423959997536, 8.26992374372849677667251547132, 8.72824685753687373233013136632, 9.21408957164806967254480733322, 10.19373572264891423372823531592, 10.734773299734977311031507967684, 11.2950126763668554635413864725, 12.11353319382241398623991324060, 12.867612631349426457432240352196, 13.36763700383526208985183361110, 14.093404687689523456033510473720, 14.91794907434701038980463795457, 15.72979099760842376402819336276, 16.60156232513108294995060808509, 17.21117973253664859615047542607, 17.788363549971999548559115501055, 18.31212373265598605764944650311