| L(s) = 1 | − 2-s + 8·3-s − 7·4-s − 8·6-s − 7·7-s + 15·8-s + 37·9-s + 12·11-s − 56·12-s + 78·13-s + 7·14-s + 41·16-s + 94·17-s − 37·18-s + 40·19-s − 56·21-s − 12·22-s − 32·23-s + 120·24-s − 78·26-s + 80·27-s + 49·28-s − 50·29-s − 248·31-s − 161·32-s + 96·33-s − 94·34-s + ⋯ |
| L(s) = 1 | − 0.353·2-s + 1.53·3-s − 7/8·4-s − 0.544·6-s − 0.377·7-s + 0.662·8-s + 1.37·9-s + 0.328·11-s − 1.34·12-s + 1.66·13-s + 0.133·14-s + 0.640·16-s + 1.34·17-s − 0.484·18-s + 0.482·19-s − 0.581·21-s − 0.116·22-s − 0.290·23-s + 1.02·24-s − 0.588·26-s + 0.570·27-s + 0.330·28-s − 0.320·29-s − 1.43·31-s − 0.889·32-s + 0.506·33-s − 0.474·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.095444940\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.095444940\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 + p T \) |
| good | 2 | \( 1 + T + p^{3} T^{2} \) |
| 3 | \( 1 - 8 T + p^{3} T^{2} \) |
| 11 | \( 1 - 12 T + p^{3} T^{2} \) |
| 13 | \( 1 - 6 p T + p^{3} T^{2} \) |
| 17 | \( 1 - 94 T + p^{3} T^{2} \) |
| 19 | \( 1 - 40 T + p^{3} T^{2} \) |
| 23 | \( 1 + 32 T + p^{3} T^{2} \) |
| 29 | \( 1 + 50 T + p^{3} T^{2} \) |
| 31 | \( 1 + 8 p T + p^{3} T^{2} \) |
| 37 | \( 1 - 434 T + p^{3} T^{2} \) |
| 41 | \( 1 - 402 T + p^{3} T^{2} \) |
| 43 | \( 1 - 68 T + p^{3} T^{2} \) |
| 47 | \( 1 + 536 T + p^{3} T^{2} \) |
| 53 | \( 1 + 22 T + p^{3} T^{2} \) |
| 59 | \( 1 + 560 T + p^{3} T^{2} \) |
| 61 | \( 1 + 278 T + p^{3} T^{2} \) |
| 67 | \( 1 - 164 T + p^{3} T^{2} \) |
| 71 | \( 1 - 672 T + p^{3} T^{2} \) |
| 73 | \( 1 + 82 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1000 T + p^{3} T^{2} \) |
| 83 | \( 1 - 448 T + p^{3} T^{2} \) |
| 89 | \( 1 + 870 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1026 T + p^{3} T^{2} \) |
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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
−12.69551318958635373038341377079, −11.03883557541485018602384750904, −9.694672046690131244583399120674, −9.258248143118897162077758202479, −8.268671407679627604996565551323, −7.60368248294959681558600152727, −5.85222442009799353217035975865, −4.04989709957727718247483837114, −3.25436067910591054760461208736, −1.30528193135988779285210397147,
1.30528193135988779285210397147, 3.25436067910591054760461208736, 4.04989709957727718247483837114, 5.85222442009799353217035975865, 7.60368248294959681558600152727, 8.268671407679627604996565551323, 9.258248143118897162077758202479, 9.694672046690131244583399120674, 11.03883557541485018602384750904, 12.69551318958635373038341377079
