| L(s) = 1 | − 9·3-s + 94·5-s + 49·7-s + 81·9-s − 52·11-s − 770·13-s − 846·15-s − 2.02e3·17-s − 1.73e3·19-s − 441·21-s + 576·23-s + 5.71e3·25-s − 729·27-s + 5.51e3·29-s − 6.33e3·31-s + 468·33-s + 4.60e3·35-s − 7.33e3·37-s + 6.93e3·39-s − 3.26e3·41-s − 5.42e3·43-s + 7.61e3·45-s − 864·47-s + 2.40e3·49-s + 1.81e4·51-s + 4.18e3·53-s − 4.88e3·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.68·5-s + 0.377·7-s + 1/3·9-s − 0.129·11-s − 1.26·13-s − 0.970·15-s − 1.69·17-s − 1.10·19-s − 0.218·21-s + 0.227·23-s + 1.82·25-s − 0.192·27-s + 1.21·29-s − 1.18·31-s + 0.0748·33-s + 0.635·35-s − 0.881·37-s + 0.729·39-s − 0.303·41-s − 0.447·43-s + 0.560·45-s − 0.0570·47-s + 1/7·49-s + 0.979·51-s + 0.204·53-s − 0.217·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + p^{2} T \) |
| 7 | \( 1 - p^{2} T \) |
| good | 5 | \( 1 - 94 T + p^{5} T^{2} \) |
| 11 | \( 1 + 52 T + p^{5} T^{2} \) |
| 13 | \( 1 + 770 T + p^{5} T^{2} \) |
| 17 | \( 1 + 2022 T + p^{5} T^{2} \) |
| 19 | \( 1 + 1732 T + p^{5} T^{2} \) |
| 23 | \( 1 - 576 T + p^{5} T^{2} \) |
| 29 | \( 1 - 5518 T + p^{5} T^{2} \) |
| 31 | \( 1 + 6336 T + p^{5} T^{2} \) |
| 37 | \( 1 + 7338 T + p^{5} T^{2} \) |
| 41 | \( 1 + 3262 T + p^{5} T^{2} \) |
| 43 | \( 1 + 5420 T + p^{5} T^{2} \) |
| 47 | \( 1 + 864 T + p^{5} T^{2} \) |
| 53 | \( 1 - 4182 T + p^{5} T^{2} \) |
| 59 | \( 1 - 11220 T + p^{5} T^{2} \) |
| 61 | \( 1 + 45602 T + p^{5} T^{2} \) |
| 67 | \( 1 + 1396 T + p^{5} T^{2} \) |
| 71 | \( 1 + 18720 T + p^{5} T^{2} \) |
| 73 | \( 1 - 46362 T + p^{5} T^{2} \) |
| 79 | \( 1 + 97424 T + p^{5} T^{2} \) |
| 83 | \( 1 - 81228 T + p^{5} T^{2} \) |
| 89 | \( 1 + 3182 T + p^{5} T^{2} \) |
| 97 | \( 1 - 4914 T + p^{5} T^{2} \) |
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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
−10.35749228705582220295152078807, −9.437246066015093430412954224879, −8.600208057593727631261552046511, −7.03946090668062767800773468600, −6.34704702352499088596478392734, −5.28815891380267984936348559771, −4.55327031563866517195665135458, −2.49554993039318061286789063044, −1.71341071650914582423625339604, 0,
1.71341071650914582423625339604, 2.49554993039318061286789063044, 4.55327031563866517195665135458, 5.28815891380267984936348559771, 6.34704702352499088596478392734, 7.03946090668062767800773468600, 8.600208057593727631261552046511, 9.437246066015093430412954224879, 10.35749228705582220295152078807
