| L(s) = 1 | − 9·3-s − 34·5-s + 49·7-s + 81·9-s + 332·11-s − 1.02e3·13-s + 306·15-s + 922·17-s − 452·19-s − 441·21-s + 3.77e3·23-s − 1.96e3·25-s − 729·27-s + 1.16e3·29-s + 9.79e3·31-s − 2.98e3·33-s − 1.66e3·35-s + 2.39e3·37-s + 9.23e3·39-s − 7.23e3·41-s − 4.65e3·43-s − 2.75e3·45-s − 2.46e4·47-s + 2.40e3·49-s − 8.29e3·51-s + 1.11e3·53-s − 1.12e4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.608·5-s + 0.377·7-s + 1/3·9-s + 0.827·11-s − 1.68·13-s + 0.351·15-s + 0.773·17-s − 0.287·19-s − 0.218·21-s + 1.48·23-s − 0.630·25-s − 0.192·27-s + 0.257·29-s + 1.83·31-s − 0.477·33-s − 0.229·35-s + 0.287·37-s + 0.972·39-s − 0.671·41-s − 0.383·43-s − 0.202·45-s − 1.62·47-s + 1/7·49-s − 0.446·51-s + 0.0542·53-s − 0.503·55-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 | \( 1 + 34 T + p^{5} T^{2} \) |
| 11 | \( 1 - 332 T + p^{5} T^{2} \) |
| 13 | \( 1 + 1026 T + p^{5} T^{2} \) |
| 17 | \( 1 - 922 T + p^{5} T^{2} \) |
| 19 | \( 1 + 452 T + p^{5} T^{2} \) |
| 23 | \( 1 - 3776 T + p^{5} T^{2} \) |
| 29 | \( 1 - 1166 T + p^{5} T^{2} \) |
| 31 | \( 1 - 9792 T + p^{5} T^{2} \) |
| 37 | \( 1 - 2390 T + p^{5} T^{2} \) |
| 41 | \( 1 + 7230 T + p^{5} T^{2} \) |
| 43 | \( 1 + 4652 T + p^{5} T^{2} \) |
| 47 | \( 1 + 24672 T + p^{5} T^{2} \) |
| 53 | \( 1 - 1110 T + p^{5} T^{2} \) |
| 59 | \( 1 + 46892 T + p^{5} T^{2} \) |
| 61 | \( 1 + 9762 T + p^{5} T^{2} \) |
| 67 | \( 1 - 26252 T + p^{5} T^{2} \) |
| 71 | \( 1 + 65440 T + p^{5} T^{2} \) |
| 73 | \( 1 + 5606 T + p^{5} T^{2} \) |
| 79 | \( 1 - 9840 T + p^{5} T^{2} \) |
| 83 | \( 1 + 61108 T + p^{5} T^{2} \) |
| 89 | \( 1 + 62958 T + p^{5} T^{2} \) |
| 97 | \( 1 + 37838 T + p^{5} T^{2} \) |
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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
−10.25856883851818674074313788294, −9.487309125731153278184453242928, −8.231654937444945741664297685900, −7.34548424296612183582124217320, −6.45795571312658408086281263883, −5.11008873504651754148495748494, −4.38399259143924650933903289331, −2.95600532809480101128262969901, −1.32791277727057133022615217788, 0,
1.32791277727057133022615217788, 2.95600532809480101128262969901, 4.38399259143924650933903289331, 5.11008873504651754148495748494, 6.45795571312658408086281263883, 7.34548424296612183582124217320, 8.231654937444945741664297685900, 9.487309125731153278184453242928, 10.25856883851818674074313788294
