| L(s) = 1 | + 5.04·2-s + 2.60·3-s + 17.4·4-s + 13.4·5-s + 13.1·6-s + 22.2·7-s + 47.4·8-s − 20.1·9-s + 67.6·10-s − 33.6·11-s + 45.4·12-s − 73.4·13-s + 111.·14-s + 35.0·15-s + 100.·16-s − 101.·18-s − 42.5·19-s + 233.·20-s + 57.9·21-s − 169.·22-s − 59.2·23-s + 123.·24-s + 55.0·25-s − 370.·26-s − 123.·27-s + 386.·28-s − 21.3·29-s + ⋯ |
| L(s) = 1 | + 1.78·2-s + 0.502·3-s + 2.17·4-s + 1.20·5-s + 0.895·6-s + 1.19·7-s + 2.09·8-s − 0.747·9-s + 2.13·10-s − 0.922·11-s + 1.09·12-s − 1.56·13-s + 2.13·14-s + 0.602·15-s + 1.56·16-s − 1.33·18-s − 0.513·19-s + 2.61·20-s + 0.602·21-s − 1.64·22-s − 0.537·23-s + 1.05·24-s + 0.440·25-s − 2.79·26-s − 0.877·27-s + 2.61·28-s − 0.136·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.998513334\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.998513334\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 - 5.04T + 8T^{2} \) |
| 3 | \( 1 - 2.60T + 27T^{2} \) |
| 5 | \( 1 - 13.4T + 125T^{2} \) |
| 7 | \( 1 - 22.2T + 343T^{2} \) |
| 11 | \( 1 + 33.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 73.4T + 2.19e3T^{2} \) |
| 19 | \( 1 + 42.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 59.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + 21.3T + 2.43e4T^{2} \) |
| 31 | \( 1 - 42.2T + 2.97e4T^{2} \) |
| 37 | \( 1 - 265.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 80.0T + 6.89e4T^{2} \) |
| 43 | \( 1 - 353.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 52.4T + 1.03e5T^{2} \) |
| 53 | \( 1 - 551.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 508.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 671.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 859.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 147.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 522.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 245.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 293.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 72.0T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.38e3T + 9.12e5T^{2} \) |
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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
−11.63064754165400535426180635274, −10.73779562079618695471431326512, −9.674778296346418901455651318270, −8.235963758049758909855201282864, −7.25392217499869294897228868819, −5.83688354797005032744661284908, −5.30653330535734058924376189113, −4.32715981421542794272048262291, −2.57585420925917948519762422843, −2.20354503888185315116604074935,
2.20354503888185315116604074935, 2.57585420925917948519762422843, 4.32715981421542794272048262291, 5.30653330535734058924376189113, 5.83688354797005032744661284908, 7.25392217499869294897228868819, 8.235963758049758909855201282864, 9.674778296346418901455651318270, 10.73779562079618695471431326512, 11.63064754165400535426180635274
