| L(s) = 1 | − 43.0·2-s + 38.5·3-s + 1.34e3·4-s + 1.11e3·5-s − 1.66e3·6-s + 1.22e4·7-s − 3.58e4·8-s − 1.81e4·9-s − 4.81e4·10-s − 6.87e4·11-s + 5.18e4·12-s + 5.05e4·13-s − 5.28e5·14-s + 4.31e4·15-s + 8.58e5·16-s + 7.84e5·18-s + 2.03e5·19-s + 1.50e6·20-s + 4.73e5·21-s + 2.96e6·22-s + 7.01e5·23-s − 1.38e6·24-s − 7.03e5·25-s − 2.17e6·26-s − 1.46e6·27-s + 1.65e7·28-s + 1.57e6·29-s + ⋯ |
| L(s) = 1 | − 1.90·2-s + 0.274·3-s + 2.62·4-s + 0.800·5-s − 0.523·6-s + 1.93·7-s − 3.09·8-s − 0.924·9-s − 1.52·10-s − 1.41·11-s + 0.721·12-s + 0.491·13-s − 3.67·14-s + 0.219·15-s + 3.27·16-s + 1.76·18-s + 0.357·19-s + 2.10·20-s + 0.530·21-s + 2.69·22-s + 0.522·23-s − 0.851·24-s − 0.359·25-s − 0.935·26-s − 0.528·27-s + 5.07·28-s + 0.412·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 + 43.0T + 512T^{2} \) |
| 3 | \( 1 - 38.5T + 1.96e4T^{2} \) |
| 5 | \( 1 - 1.11e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 1.22e4T + 4.03e7T^{2} \) |
| 11 | \( 1 + 6.87e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 5.05e4T + 1.06e10T^{2} \) |
| 19 | \( 1 - 2.03e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 7.01e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 1.57e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 5.39e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 6.75e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 3.38e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.78e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 6.37e5T + 1.11e15T^{2} \) |
| 53 | \( 1 + 8.03e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.59e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 8.02e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.11e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 7.89e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 8.42e7T + 5.88e16T^{2} \) |
| 79 | \( 1 + 4.78e7T + 1.19e17T^{2} \) |
| 83 | \( 1 + 2.08e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 4.13e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 7.40e8T + 7.60e17T^{2} \) |
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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
−9.587081888929928841704964095267, −8.701178278266746102958728889775, −8.056412402217016114352618146697, −7.50062519053509491426519143708, −5.99496063768860987178303568417, −5.15149619170184432059759740983, −2.87085743712586247242263375828, −2.02015710262775647275972115700, −1.28808712077025124509760254430, 0,
1.28808712077025124509760254430, 2.02015710262775647275972115700, 2.87085743712586247242263375828, 5.15149619170184432059759740983, 5.99496063768860987178303568417, 7.50062519053509491426519143708, 8.056412402217016114352618146697, 8.701178278266746102958728889775, 9.587081888929928841704964095267
