L(s) = 1 | + (0.292 − 1.70i)3-s + 3.41i·5-s + i·7-s + (−2.82 − i)9-s + 2·11-s − 3.41·13-s + (5.82 + i)15-s − 4.82i·17-s + 6.24i·19-s + (1.70 + 0.292i)21-s + 4·23-s − 6.65·25-s + (−2.53 + 4.53i)27-s + 4.82i·29-s + 1.17i·31-s + ⋯ |
L(s) = 1 | + (0.169 − 0.985i)3-s + 1.52i·5-s + 0.377i·7-s + (−0.942 − 0.333i)9-s + 0.603·11-s − 0.946·13-s + (1.50 + 0.258i)15-s − 1.17i·17-s + 1.43i·19-s + (0.372 + 0.0639i)21-s + 0.834·23-s − 1.33·25-s + (−0.487 + 0.872i)27-s + 0.896i·29-s + 0.210i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.169 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.169 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.287773060\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.287773060\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.292 + 1.70i)T \) |
| 7 | \( 1 - iT \) |
good | 5 | \( 1 - 3.41iT - 5T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 3.41T + 13T^{2} \) |
| 17 | \( 1 + 4.82iT - 17T^{2} \) |
| 19 | \( 1 - 6.24iT - 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 4.82iT - 29T^{2} \) |
| 31 | \( 1 - 1.17iT - 31T^{2} \) |
| 37 | \( 1 - 0.828T + 37T^{2} \) |
| 41 | \( 1 - 10iT - 41T^{2} \) |
| 43 | \( 1 - 10.4iT - 43T^{2} \) |
| 47 | \( 1 + 1.65T + 47T^{2} \) |
| 53 | \( 1 - 13.3iT - 53T^{2} \) |
| 59 | \( 1 + 6.24T + 59T^{2} \) |
| 61 | \( 1 + 14.2T + 61T^{2} \) |
| 67 | \( 1 + 9.31iT - 67T^{2} \) |
| 71 | \( 1 - 0.343T + 71T^{2} \) |
| 73 | \( 1 + 3.17T + 73T^{2} \) |
| 79 | \( 1 - 4iT - 79T^{2} \) |
| 83 | \( 1 - 11.4T + 83T^{2} \) |
| 89 | \( 1 + 7.65iT - 89T^{2} \) |
| 97 | \( 1 - 5.31T + 97T^{2} \) |
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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
−9.698893568269443911513108680522, −9.048904483055430864790915260879, −7.78977475497575906008998584165, −7.41248801500918505953611252684, −6.54258487199888575892367531678, −6.04084610456582072696656467202, −4.79473346019522490811489394556, −3.18404811600767054794239925937, −2.81367981873110252533057059980, −1.54866066467328557644005358046,
0.51467919315431230391340596404, 2.11151526355999967497326615864, 3.56851853061187291201465442175, 4.44941985385296528963664656577, 4.95037152150718178641442715631, 5.82815182417317153280053331089, 7.05346428873661180380194896735, 8.097332058694949082955078103708, 8.890705555106794527143606883402, 9.240641505610054832842889971021