| L(s) = 1 | − 2-s + 3-s − 4-s − 6-s + 7-s + 3·8-s + 9-s − 12-s + 6·13-s − 14-s − 16-s + 2·17-s − 18-s − 4·19-s + 21-s + 3·24-s − 6·26-s + 27-s − 28-s + 2·29-s + 8·31-s − 5·32-s − 2·34-s − 36-s − 6·37-s + 4·38-s + 6·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.408·6-s + 0.377·7-s + 1.06·8-s + 1/3·9-s − 0.288·12-s + 1.66·13-s − 0.267·14-s − 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.917·19-s + 0.218·21-s + 0.612·24-s − 1.17·26-s + 0.192·27-s − 0.188·28-s + 0.371·29-s + 1.43·31-s − 0.883·32-s − 0.342·34-s − 1/6·36-s − 0.986·37-s + 0.648·38-s + 0.960·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.132463656\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.132463656\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.26189372277564, −13.66289229110031, −13.38091797802903, −12.94318175322065, −12.19603936752720, −11.70234628051949, −11.02404778796694, −10.48509119763851, −10.21366571053964, −9.602548796626016, −8.929499305709319, −8.582400670605382, −8.144968901476265, −7.950134593650501, −6.886780432685989, −6.672570346919307, −5.809682209674020, −5.166761254784251, −4.590378920767033, −3.881092952220018, −3.583583708273784, −2.688870968314585, −1.822725792999675, −1.302213481603797, −0.5917745804792084,
0.5917745804792084, 1.302213481603797, 1.822725792999675, 2.688870968314585, 3.583583708273784, 3.881092952220018, 4.590378920767033, 5.166761254784251, 5.809682209674020, 6.672570346919307, 6.886780432685989, 7.950134593650501, 8.144968901476265, 8.582400670605382, 8.929499305709319, 9.602548796626016, 10.21366571053964, 10.48509119763851, 11.02404778796694, 11.70234628051949, 12.19603936752720, 12.94318175322065, 13.38091797802903, 13.66289229110031, 14.26189372277564
