| L(s) = 1 | + 4·2-s + 9·3-s + 16·4-s − 61.0·5-s + 36·6-s + 64·8-s + 81·9-s − 244.·10-s − 36.5·11-s + 144·12-s + 34.5·13-s − 549.·15-s + 256·16-s − 2.06e3·17-s + 324·18-s + 452.·19-s − 977.·20-s − 146.·22-s + 1.68e3·23-s + 576·24-s + 604.·25-s + 138.·26-s + 729·27-s − 4.76e3·29-s − 2.19e3·30-s − 5.26e3·31-s + 1.02e3·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.09·5-s + 0.408·6-s + 0.353·8-s + 0.333·9-s − 0.772·10-s − 0.0910·11-s + 0.288·12-s + 0.0567·13-s − 0.630·15-s + 0.250·16-s − 1.72·17-s + 0.235·18-s + 0.287·19-s − 0.546·20-s − 0.0643·22-s + 0.663·23-s + 0.204·24-s + 0.193·25-s + 0.0401·26-s + 0.192·27-s − 1.05·29-s − 0.446·30-s − 0.983·31-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 | \( 1 + 61.0T + 3.12e3T^{2} \) |
| 11 | \( 1 + 36.5T + 1.61e5T^{2} \) |
| 13 | \( 1 - 34.5T + 3.71e5T^{2} \) |
| 17 | \( 1 + 2.06e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 452.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.68e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.76e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.26e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.28e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 7.12e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.11e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.34e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 7.03e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.42e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.93e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.09e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 7.98e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.70e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.20e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.31e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.14e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.27e5T + 8.58e9T^{2} \) |
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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
−10.84020485050575768882622092704, −9.379939722301625998558016667219, −8.433418661647511944915712787030, −7.45647517156981617631874505979, −6.66529723581969905345452290922, −5.14032567574772662200908184455, −4.10284479977844316988881523722, −3.26979572967773874518544459932, −1.90902669746424570415111784594, 0,
1.90902669746424570415111784594, 3.26979572967773874518544459932, 4.10284479977844316988881523722, 5.14032567574772662200908184455, 6.66529723581969905345452290922, 7.45647517156981617631874505979, 8.433418661647511944915712787030, 9.379939722301625998558016667219, 10.84020485050575768882622092704
