| L(s) = 1 | − 2-s + 1.63·3-s + 4-s − 1.63·6-s + 7-s − 8-s − 0.336·9-s + 11-s + 1.63·12-s + 2.35·13-s − 14-s + 16-s − 5.69·17-s + 0.336·18-s − 5.40·19-s + 1.63·21-s − 22-s + 4.65·23-s − 1.63·24-s − 2.35·26-s − 5.44·27-s + 28-s + 1.38·29-s + 5.33·31-s − 32-s + 1.63·33-s + 5.69·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.942·3-s + 0.5·4-s − 0.666·6-s + 0.377·7-s − 0.353·8-s − 0.112·9-s + 0.301·11-s + 0.471·12-s + 0.653·13-s − 0.267·14-s + 0.250·16-s − 1.38·17-s + 0.0792·18-s − 1.23·19-s + 0.356·21-s − 0.213·22-s + 0.969·23-s − 0.333·24-s − 0.461·26-s − 1.04·27-s + 0.188·28-s + 0.257·29-s + 0.958·31-s − 0.176·32-s + 0.284·33-s + 0.977·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.914359779\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.914359779\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 - 1.63T + 3T^{2} \) |
| 13 | \( 1 - 2.35T + 13T^{2} \) |
| 17 | \( 1 + 5.69T + 17T^{2} \) |
| 19 | \( 1 + 5.40T + 19T^{2} \) |
| 23 | \( 1 - 4.65T + 23T^{2} \) |
| 29 | \( 1 - 1.38T + 29T^{2} \) |
| 31 | \( 1 - 5.33T + 31T^{2} \) |
| 37 | \( 1 - 1.04T + 37T^{2} \) |
| 41 | \( 1 - 9.98T + 41T^{2} \) |
| 43 | \( 1 - 0.265T + 43T^{2} \) |
| 47 | \( 1 - 9.07T + 47T^{2} \) |
| 53 | \( 1 - 8.31T + 53T^{2} \) |
| 59 | \( 1 - 6.58T + 59T^{2} \) |
| 61 | \( 1 - 8.71T + 61T^{2} \) |
| 67 | \( 1 - 2.43T + 67T^{2} \) |
| 71 | \( 1 - 5.66T + 71T^{2} \) |
| 73 | \( 1 + 6.43T + 73T^{2} \) |
| 79 | \( 1 - 11.7T + 79T^{2} \) |
| 83 | \( 1 + 2.28T + 83T^{2} \) |
| 89 | \( 1 - 3.22T + 89T^{2} \) |
| 97 | \( 1 + 15.8T + 97T^{2} \) |
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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
−8.654057178528715794381401521671, −8.048183549039615626096546420289, −7.14013954771056711125454203383, −6.49117565269455949018397968560, −5.68119027185315723183415592078, −4.47398646411989403949762430479, −3.79581210617063579033480756225, −2.62746907257666123300094164993, −2.15669090026787157162105975581, −0.848476187745743600657683408204,
0.848476187745743600657683408204, 2.15669090026787157162105975581, 2.62746907257666123300094164993, 3.79581210617063579033480756225, 4.47398646411989403949762430479, 5.68119027185315723183415592078, 6.49117565269455949018397968560, 7.14013954771056711125454203383, 8.048183549039615626096546420289, 8.654057178528715794381401521671
