L(s) = 1 | + (−11.1 − 1.29i)5-s + 16.2i·7-s − 40.2·11-s − 19.7i·13-s − 83.0i·17-s − 48.8·19-s − 1.61i·23-s + (121. + 28.8i)25-s − 24.5·29-s + 12.4·31-s + (21.0 − 180i)35-s + 325. i·37-s + 242.·41-s + 367. i·43-s − 204. i·47-s + ⋯ |
L(s) = 1 | + (−0.993 − 0.116i)5-s + 0.875i·7-s − 1.10·11-s − 0.422i·13-s − 1.18i·17-s − 0.589·19-s − 0.0146i·23-s + (0.973 + 0.230i)25-s − 0.157·29-s + 0.0719·31-s + (0.101 − 0.869i)35-s + 1.44i·37-s + 0.923·41-s + 1.30i·43-s − 0.634i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.116i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.993 + 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.152497587\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.152497587\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (11.1 + 1.29i)T \) |
good | 7 | \( 1 - 16.2iT - 343T^{2} \) |
| 11 | \( 1 + 40.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 19.7iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 83.0iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 48.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 1.61iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 24.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 12.4T + 2.97e4T^{2} \) |
| 37 | \( 1 - 325. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 242.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 367. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 204. iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 61.5iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 112.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 477.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 558. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 558.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.01e3iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 1.15e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.15e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 96.9T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.15e3iT - 9.12e5T^{2} \) |
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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
−10.00088695171240247143042115113, −9.028559137295051060005459865619, −8.183962045386672191228387611063, −7.60585152766906665037572370727, −6.50241972965961002212713433313, −5.32839299599788793662937447165, −4.65462759898636209791835492958, −3.28430106088384710439753590585, −2.41318541054096188250555171794, −0.55258587749737921052581322302,
0.64100420281826216527093165951, 2.29593398534495789170469447916, 3.71033914881123060810941537211, 4.28093030713821069968264573978, 5.49485973972768383863281247451, 6.69424388271704429931121888263, 7.51954984049168046884325541718, 8.137392370390727704203225957035, 9.055211770357389017011995991807, 10.40841846522510466958876703423