| L(s) = 1 | − 3-s − 3·7-s + 9-s + 4·13-s + 17-s − 2·19-s + 3·21-s − 6·23-s − 5·25-s − 27-s + 9·29-s + 4·31-s + 2·37-s − 4·39-s + 41-s + 6·43-s + 9·47-s + 2·49-s − 51-s + 13·53-s + 2·57-s + 15·59-s + 14·61-s − 3·63-s + 9·67-s + 6·69-s + 14·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.13·7-s + 1/3·9-s + 1.10·13-s + 0.242·17-s − 0.458·19-s + 0.654·21-s − 1.25·23-s − 25-s − 0.192·27-s + 1.67·29-s + 0.718·31-s + 0.328·37-s − 0.640·39-s + 0.156·41-s + 0.914·43-s + 1.31·47-s + 2/7·49-s − 0.140·51-s + 1.78·53-s + 0.264·57-s + 1.95·59-s + 1.79·61-s − 0.377·63-s + 1.09·67-s + 0.722·69-s + 1.66·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.264746412\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.264746412\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 13 T + p T^{2} \) |
| 59 | \( 1 - 15 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 13 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.77700726220303, −13.22612463387788, −12.79834665163988, −12.23951411909912, −11.88070370116257, −11.40743690295170, −10.69804544885854, −10.37433489624886, −9.761615325989992, −9.604692981182807, −8.648105290876937, −8.378948684237300, −7.786852983975622, −7.045993994091761, −6.460906583673448, −6.296500485860186, −5.656499168205172, −5.222573929350920, −4.250338892620122, −3.896173814579173, −3.481630857112473, −2.446082432118020, −2.170767781277940, −0.8141243007608844, −0.7146308153660612,
0.7146308153660612, 0.8141243007608844, 2.170767781277940, 2.446082432118020, 3.481630857112473, 3.896173814579173, 4.250338892620122, 5.222573929350920, 5.656499168205172, 6.296500485860186, 6.460906583673448, 7.045993994091761, 7.786852983975622, 8.378948684237300, 8.648105290876937, 9.604692981182807, 9.761615325989992, 10.37433489624886, 10.69804544885854, 11.40743690295170, 11.88070370116257, 12.23951411909912, 12.79834665163988, 13.22612463387788, 13.77700726220303
