| L(s) = 1 | + 4.82·2-s + 4.44·3-s + 15.2·4-s − 12.7·5-s + 21.4·6-s + 26.1·7-s + 35.1·8-s − 7.25·9-s − 61.6·10-s + 42.3·11-s + 67.9·12-s + 126.·14-s − 56.7·15-s + 47.3·16-s − 27.3·17-s − 35.0·18-s − 13.1·19-s − 195.·20-s + 116.·21-s + 204.·22-s − 28.7·23-s + 156.·24-s + 38.1·25-s − 152.·27-s + 400.·28-s − 141.·29-s − 273.·30-s + ⋯ |
| L(s) = 1 | + 1.70·2-s + 0.855·3-s + 1.91·4-s − 1.14·5-s + 1.45·6-s + 1.41·7-s + 1.55·8-s − 0.268·9-s − 1.94·10-s + 1.16·11-s + 1.63·12-s + 2.41·14-s − 0.976·15-s + 0.740·16-s − 0.389·17-s − 0.458·18-s − 0.158·19-s − 2.18·20-s + 1.20·21-s + 1.98·22-s − 0.261·23-s + 1.32·24-s + 0.304·25-s − 1.08·27-s + 2.70·28-s − 0.906·29-s − 1.66·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.208365494\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.208365494\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 \) |
| good | 2 | \( 1 - 4.82T + 8T^{2} \) |
| 3 | \( 1 - 4.44T + 27T^{2} \) |
| 5 | \( 1 + 12.7T + 125T^{2} \) |
| 7 | \( 1 - 26.1T + 343T^{2} \) |
| 11 | \( 1 - 42.3T + 1.33e3T^{2} \) |
| 17 | \( 1 + 27.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 13.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 28.7T + 1.21e4T^{2} \) |
| 29 | \( 1 + 141.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 56.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + 313.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 352.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 320.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 339.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 349.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 258.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 650.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 894.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 741.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 820.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 199.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 541.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 380.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.43e3T + 9.12e5T^{2} \) |
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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
−12.29207079966111817965988349014, −11.59194712713364567501054575357, −11.04756527174274924253228853678, −8.945564490093067974440049787386, −8.002074947835165324368126467007, −7.00026535325092248915987749129, −5.48454239493736421282434731845, −4.24294557926715281329569859220, −3.62124467118151988915501524518, −2.06083530624351401600635363492,
2.06083530624351401600635363492, 3.62124467118151988915501524518, 4.24294557926715281329569859220, 5.48454239493736421282434731845, 7.00026535325092248915987749129, 8.002074947835165324368126467007, 8.945564490093067974440049787386, 11.04756527174274924253228853678, 11.59194712713364567501054575357, 12.29207079966111817965988349014
