L(s) = 1 | + 2.67i·2-s − 5.13·4-s − 3.20·5-s − 8.37i·8-s − 8.55i·10-s − 4.06i·11-s + 2.99i·13-s + 12.0·16-s − 2.54·17-s − 0.165i·19-s + 16.4·20-s + 10.8·22-s − 4.04i·23-s + 5.26·25-s − 8.00·26-s + ⋯ |
L(s) = 1 | + 1.88i·2-s − 2.56·4-s − 1.43·5-s − 2.95i·8-s − 2.70i·10-s − 1.22i·11-s + 0.831i·13-s + 3.02·16-s − 0.616·17-s − 0.0379i·19-s + 3.67·20-s + 2.31·22-s − 0.843i·23-s + 1.05·25-s − 1.57·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.156 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.156 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7669331320\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7669331320\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 2.67iT - 2T^{2} \) |
| 5 | \( 1 + 3.20T + 5T^{2} \) |
| 11 | \( 1 + 4.06iT - 11T^{2} \) |
| 13 | \( 1 - 2.99iT - 13T^{2} \) |
| 17 | \( 1 + 2.54T + 17T^{2} \) |
| 19 | \( 1 + 0.165iT - 19T^{2} \) |
| 23 | \( 1 + 4.04iT - 23T^{2} \) |
| 29 | \( 1 - 2.12iT - 29T^{2} \) |
| 31 | \( 1 - 7.70iT - 31T^{2} \) |
| 37 | \( 1 - 11.8T + 37T^{2} \) |
| 41 | \( 1 - 0.629T + 41T^{2} \) |
| 43 | \( 1 - 3.94T + 43T^{2} \) |
| 47 | \( 1 - 5.45T + 47T^{2} \) |
| 53 | \( 1 - 2.48iT - 53T^{2} \) |
| 59 | \( 1 + 8.84T + 59T^{2} \) |
| 61 | \( 1 + 10.8iT - 61T^{2} \) |
| 67 | \( 1 + 1.14T + 67T^{2} \) |
| 71 | \( 1 + 9.33iT - 71T^{2} \) |
| 73 | \( 1 + 6.10iT - 73T^{2} \) |
| 79 | \( 1 + 12.8T + 79T^{2} \) |
| 83 | \( 1 + 3.76T + 83T^{2} \) |
| 89 | \( 1 - 18.1T + 89T^{2} \) |
| 97 | \( 1 - 12.1iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.195775007575773846463631940749, −8.780339235489572570874347334828, −8.040050043542936515623962584180, −7.45458339104809788433528384108, −6.60670908254369186942698216872, −6.00738308967946573862851897078, −4.78283931529170586722821625904, −4.25544700960560058955640008180, −3.28084862795638739920099752285, −0.54204414652845924332609013735,
0.74186014754067278384561553977, 2.20205962663978427758408780616, 3.12660039706298085413265284984, 4.24188740199578132928561584238, 4.39155750167568545376143660291, 5.71598737542719326592261005623, 7.42030395178515548564867406493, 7.915371184277763823991461945602, 8.883014242352548372119838476256, 9.675873153651137814438142167366