| L(s) = 1 | + 4·2-s + 2·3-s + 8·4-s + 17·5-s + 8·6-s + 20·7-s − 23·9-s + 68·10-s − 32·11-s + 16·12-s + 80·14-s + 34·15-s − 64·16-s − 13·17-s − 92·18-s + 30·19-s + 136·20-s + 40·21-s − 128·22-s + 78·23-s + 164·25-s − 100·27-s + 160·28-s + 197·29-s + 136·30-s − 74·31-s − 256·32-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.384·3-s + 4-s + 1.52·5-s + 0.544·6-s + 1.07·7-s − 0.851·9-s + 2.15·10-s − 0.877·11-s + 0.384·12-s + 1.52·14-s + 0.585·15-s − 16-s − 0.185·17-s − 1.20·18-s + 0.362·19-s + 1.52·20-s + 0.415·21-s − 1.24·22-s + 0.707·23-s + 1.31·25-s − 0.712·27-s + 1.07·28-s + 1.26·29-s + 0.827·30-s − 0.428·31-s − 1.41·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.655651845\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.655651845\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 \) |
| good | 2 | \( 1 - p^{2} T + p^{3} T^{2} \) |
| 3 | \( 1 - 2 T + p^{3} T^{2} \) |
| 5 | \( 1 - 17 T + p^{3} T^{2} \) |
| 7 | \( 1 - 20 T + p^{3} T^{2} \) |
| 11 | \( 1 + 32 T + p^{3} T^{2} \) |
| 17 | \( 1 + 13 T + p^{3} T^{2} \) |
| 19 | \( 1 - 30 T + p^{3} T^{2} \) |
| 23 | \( 1 - 78 T + p^{3} T^{2} \) |
| 29 | \( 1 - 197 T + p^{3} T^{2} \) |
| 31 | \( 1 + 74 T + p^{3} T^{2} \) |
| 37 | \( 1 + 227 T + p^{3} T^{2} \) |
| 41 | \( 1 + 165 T + p^{3} T^{2} \) |
| 43 | \( 1 + 156 T + p^{3} T^{2} \) |
| 47 | \( 1 + 162 T + p^{3} T^{2} \) |
| 53 | \( 1 - 93 T + p^{3} T^{2} \) |
| 59 | \( 1 + 864 T + p^{3} T^{2} \) |
| 61 | \( 1 - 145 T + p^{3} T^{2} \) |
| 67 | \( 1 - 862 T + p^{3} T^{2} \) |
| 71 | \( 1 - 654 T + p^{3} T^{2} \) |
| 73 | \( 1 - 215 T + p^{3} T^{2} \) |
| 79 | \( 1 + 76 T + p^{3} T^{2} \) |
| 83 | \( 1 - 628 T + p^{3} T^{2} \) |
| 89 | \( 1 + 266 T + p^{3} T^{2} \) |
| 97 | \( 1 - 238 T + p^{3} T^{2} \) |
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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
−12.66746279982993798668021635405, −11.52553212095014110749733446406, −10.55416122650222327416040455556, −9.251100198264978672655011232009, −8.218258934247478970719781141043, −6.58718769243799798800284477673, −5.42989832211821498599343285175, −4.95832285748162899043532097695, −3.08468617285947350437020469669, −2.03820440777890733485896006326,
2.03820440777890733485896006326, 3.08468617285947350437020469669, 4.95832285748162899043532097695, 5.42989832211821498599343285175, 6.58718769243799798800284477673, 8.218258934247478970719781141043, 9.251100198264978672655011232009, 10.55416122650222327416040455556, 11.52553212095014110749733446406, 12.66746279982993798668021635405
