| L(s) = 1 | − 4·2-s − 9·3-s + 16·4-s − 4.50·5-s + 36·6-s − 64·8-s + 81·9-s + 18.0·10-s − 116.·11-s − 144·12-s + 85.4·13-s + 40.5·15-s + 256·16-s − 33.2·17-s − 324·18-s − 635.·19-s − 72.0·20-s + 464.·22-s + 2.72e3·23-s + 576·24-s − 3.10e3·25-s − 341.·26-s − 729·27-s + 5.86e3·29-s − 162.·30-s − 279.·31-s − 1.02e3·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.0805·5-s + 0.408·6-s − 0.353·8-s + 0.333·9-s + 0.0569·10-s − 0.289·11-s − 0.288·12-s + 0.140·13-s + 0.0465·15-s + 0.250·16-s − 0.0279·17-s − 0.235·18-s − 0.403·19-s − 0.0402·20-s + 0.204·22-s + 1.07·23-s + 0.204·24-s − 0.993·25-s − 0.0991·26-s − 0.192·27-s + 1.29·29-s − 0.0328·30-s − 0.0521·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 3 | \( 1 + 9T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 4.50T + 3.12e3T^{2} \) |
| 11 | \( 1 + 116.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 85.4T + 3.71e5T^{2} \) |
| 17 | \( 1 + 33.2T + 1.41e6T^{2} \) |
| 19 | \( 1 + 635.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.72e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.86e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 279.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 3.03e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 819.T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.11e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 7.40e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.36e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.23e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.26e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.23e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.02e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.69e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.35e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.05e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 9.21e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.52e5T + 8.58e9T^{2} \) |
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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
−10.51162765321645364134504111242, −9.586180665313000957081469432667, −8.573375545485203626352289805638, −7.58062582538427186700087087464, −6.60382439829072339785833393169, −5.60757472547168797879158001429, −4.33154866655228985853842867330, −2.76493769847232943658954962019, −1.27541090696420692534347657320, 0,
1.27541090696420692534347657320, 2.76493769847232943658954962019, 4.33154866655228985853842867330, 5.60757472547168797879158001429, 6.60382439829072339785833393169, 7.58062582538427186700087087464, 8.573375545485203626352289805638, 9.586180665313000957081469432667, 10.51162765321645364134504111242
