L(s) = 1 | + (0.5 − 0.866i)5-s + (0.5 − 0.866i)11-s + (0.5 − 0.866i)13-s + 17-s + 23-s + (−0.5 − 0.866i)25-s + (0.5 − 0.866i)29-s + (0.5 − 0.866i)31-s + (−0.5 − 0.866i)37-s + (0.5 + 0.866i)41-s + (−0.5 − 0.866i)43-s − 47-s + (0.5 + 0.866i)53-s + (−0.5 − 0.866i)55-s − 59-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)5-s + (0.5 − 0.866i)11-s + (0.5 − 0.866i)13-s + 17-s + 23-s + (−0.5 − 0.866i)25-s + (0.5 − 0.866i)29-s + (0.5 − 0.866i)31-s + (−0.5 − 0.866i)37-s + (0.5 + 0.866i)41-s + (−0.5 − 0.866i)43-s − 47-s + (0.5 + 0.866i)53-s + (−0.5 − 0.866i)55-s − 59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0288 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0288 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.526822667 - 1.571532071i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.526822667 - 1.571532071i\) |
\(L(1)\) |
\(\approx\) |
\(1.241813134 - 0.4471853487i\) |
\(L(1)\) |
\(\approx\) |
\(1.241813134 - 0.4471853487i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 \) |
good | 5 | \( 1 + (0.5 - 0.866i)T \) |
| 11 | \( 1 + (0.5 - 0.866i)T \) |
| 13 | \( 1 + (0.5 - 0.866i)T \) |
| 17 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + (0.5 - 0.866i)T \) |
| 31 | \( 1 + (0.5 - 0.866i)T \) |
| 37 | \( 1 + (-0.5 - 0.866i)T \) |
| 41 | \( 1 + (0.5 + 0.866i)T \) |
| 43 | \( 1 + (-0.5 - 0.866i)T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 + (0.5 + 0.866i)T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 + (0.5 + 0.866i)T \) |
| 71 | \( 1 + (0.5 + 0.866i)T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 + (-0.5 + 0.866i)T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 + (-0.5 - 0.866i)T \) |
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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
−19.14496155021566410076064093340, −18.35194279169889063884849833289, −17.81743792455859835500919756841, −17.08122352021114159370344818424, −16.4334455305118362965433625427, −15.56174312804054428147021971962, −14.76364934548583539439180637832, −14.32563448785847193397607726230, −13.68787938330022302224783133633, −12.81611459560399370887054852775, −12.03919551078433094416834631880, −11.37446249871775066239372123340, −10.57345280019030913470835956692, −9.946033491125017781087512735950, −9.28627181380539157602260438935, −8.50773191552076049732098086532, −7.48076811370854159044614526039, −6.764108407728408642963942647446, −6.40302787710754958904039385080, −5.30642057418047418352140092214, −4.615654799693267846441800147370, −3.52687169814654656495593138769, −2.98833654976795937350796044201, −1.86521111330208911414007078224, −1.2895081672691595000672806419,
0.74537437472203384643252373066, 1.23539428717525119825051702922, 2.444852758250923331034216107684, 3.31205369160048431069906680512, 4.11793845782541257555751420579, 5.07701323138485085760216538786, 5.75121584299082183597668260746, 6.249235154962535015173510617353, 7.368763550862842092426957421989, 8.279745287928362984613204833849, 8.65066209562603756431123009737, 9.583614088505039181008854428, 10.13117918165474183092463832453, 11.06305629289714055277124815953, 11.7488499836069496651264607440, 12.56014685824004920359249665280, 13.17854107160826852480069260162, 13.7803294113339383787510023560, 14.48733367526468464171731307803, 15.385030494449896363964855361209, 16.065052918470029142025406820847, 16.80509065723382990392510014694, 17.21043323492346427465026468635, 17.997132158634205858892593258057, 18.828996728318536660888371240325