| L(s) = 1 | − 2-s + 4-s − 8-s + 11-s − 13-s + 16-s − 8·17-s + 4·19-s − 22-s + 4·23-s + 26-s + 9·29-s − 8·31-s − 32-s + 8·34-s + 2·37-s − 4·38-s + 5·41-s − 5·43-s + 44-s − 4·46-s − 4·47-s − 52-s + 9·53-s − 9·58-s − 15·59-s − 10·61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s + 0.301·11-s − 0.277·13-s + 1/4·16-s − 1.94·17-s + 0.917·19-s − 0.213·22-s + 0.834·23-s + 0.196·26-s + 1.67·29-s − 1.43·31-s − 0.176·32-s + 1.37·34-s + 0.328·37-s − 0.648·38-s + 0.780·41-s − 0.762·43-s + 0.150·44-s − 0.589·46-s − 0.583·47-s − 0.138·52-s + 1.23·53-s − 1.18·58-s − 1.95·59-s − 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4426062872\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4426062872\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 11 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 15 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 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
−12.62028767347004, −12.18841871410763, −11.74284074954532, −11.22925757163876, −10.89720051462516, −10.49753438082362, −9.883601662808675, −9.434935295652811, −9.003215649327005, −8.697185858469691, −8.191647310108367, −7.529971100896754, −7.125040520501228, −6.786231653280276, −6.237904663688475, −5.720659885953348, −5.114486862719981, −4.469785681393737, −4.230775826034911, −3.305337627781935, −2.820383573734393, −2.422295400807566, −1.505732056868602, −1.267424448233722, −0.2013223196167277,
0.2013223196167277, 1.267424448233722, 1.505732056868602, 2.422295400807566, 2.820383573734393, 3.305337627781935, 4.230775826034911, 4.469785681393737, 5.114486862719981, 5.720659885953348, 6.237904663688475, 6.786231653280276, 7.125040520501228, 7.529971100896754, 8.191647310108367, 8.697185858469691, 9.003215649327005, 9.434935295652811, 9.883601662808675, 10.49753438082362, 10.89720051462516, 11.22925757163876, 11.74284074954532, 12.18841871410763, 12.62028767347004
