| L(s) = 1 | + 3.27·2-s − 3·3-s + 2.75·4-s + 5·5-s − 9.83·6-s − 33.3·7-s − 17.2·8-s + 9·9-s + 16.3·10-s − 11·11-s − 8.26·12-s − 24.2·13-s − 109.·14-s − 15·15-s − 78.4·16-s + 69.7·17-s + 29.5·18-s − 125.·19-s + 13.7·20-s + 99.9·21-s − 36.0·22-s + 130.·23-s + 51.6·24-s + 25·25-s − 79.6·26-s − 27·27-s − 91.8·28-s + ⋯ |
| L(s) = 1 | + 1.15·2-s − 0.577·3-s + 0.344·4-s + 0.447·5-s − 0.669·6-s − 1.79·7-s − 0.760·8-s + 0.333·9-s + 0.518·10-s − 0.301·11-s − 0.198·12-s − 0.518·13-s − 2.08·14-s − 0.258·15-s − 1.22·16-s + 0.994·17-s + 0.386·18-s − 1.51·19-s + 0.153·20-s + 1.03·21-s − 0.349·22-s + 1.18·23-s + 0.438·24-s + 0.200·25-s − 0.600·26-s − 0.192·27-s − 0.619·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 5 | \( 1 - 5T \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 - 3.27T + 8T^{2} \) |
| 7 | \( 1 + 33.3T + 343T^{2} \) |
| 13 | \( 1 + 24.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 69.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 125.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 130.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 238.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 133.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 166.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 297.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 463.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 585.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 40.0T + 1.48e5T^{2} \) |
| 59 | \( 1 + 312.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 391.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 858.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 583.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 368.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 438.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 877.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 9.99e2T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.30e3T + 9.12e5T^{2} \) |
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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
−12.50252854398827666183198757342, −11.01984044534980877778517120185, −9.884227708582963514601395771885, −9.102814911172772686951739322571, −7.10489995270158592949818249408, −6.13304821742037061415423369891, −5.40295365547241660136231461319, −3.97769407883411266935685157409, −2.78091319458968493639424533927, 0,
2.78091319458968493639424533927, 3.97769407883411266935685157409, 5.40295365547241660136231461319, 6.13304821742037061415423369891, 7.10489995270158592949818249408, 9.102814911172772686951739322571, 9.884227708582963514601395771885, 11.01984044534980877778517120185, 12.50252854398827666183198757342
