| L(s) = 1 | + 2·2-s + 2·4-s + 4·5-s − 5·7-s − 3·9-s + 8·10-s − 11-s + 4·13-s − 10·14-s − 4·16-s + 17-s − 6·18-s + 2·19-s + 8·20-s − 2·22-s − 2·23-s + 11·25-s + 8·26-s − 10·28-s − 3·29-s + 4·31-s − 8·32-s + 2·34-s − 20·35-s − 6·36-s − 2·37-s + 4·38-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 1.78·5-s − 1.88·7-s − 9-s + 2.52·10-s − 0.301·11-s + 1.10·13-s − 2.67·14-s − 16-s + 0.242·17-s − 1.41·18-s + 0.458·19-s + 1.78·20-s − 0.426·22-s − 0.417·23-s + 11/5·25-s + 1.56·26-s − 1.88·28-s − 0.557·29-s + 0.718·31-s − 1.41·32-s + 0.342·34-s − 3.38·35-s − 36-s − 0.328·37-s + 0.648·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.327097641\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.327097641\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−12.96675147622474839505859210517, −12.05927031115559358055334853520, −10.64823087958760524092405313569, −9.656220340411693126871827921078, −8.878162982617934095106316520221, −6.61714696112762425798852507107, −6.01118959988592874072183945220, −5.42343612788189796607524603056, −3.50251652351210255345821016437, −2.60812448202309704136989954701,
2.60812448202309704136989954701, 3.50251652351210255345821016437, 5.42343612788189796607524603056, 6.01118959988592874072183945220, 6.61714696112762425798852507107, 8.878162982617934095106316520221, 9.656220340411693126871827921078, 10.64823087958760524092405313569, 12.05927031115559358055334853520, 12.96675147622474839505859210517
