| L(s) = 1 | − 0.173·2-s + 3-s − 1.96·4-s + 5-s − 0.173·6-s − 4.18·7-s + 0.690·8-s + 9-s − 0.173·10-s + 2.34·11-s − 1.96·12-s − 13-s + 0.728·14-s + 15-s + 3.81·16-s + 2.05·17-s − 0.173·18-s − 6.58·19-s − 1.96·20-s − 4.18·21-s − 0.408·22-s + 0.591·23-s + 0.690·24-s + 25-s + 0.173·26-s + 27-s + 8.24·28-s + ⋯ |
| L(s) = 1 | − 0.122·2-s + 0.577·3-s − 0.984·4-s + 0.447·5-s − 0.0709·6-s − 1.58·7-s + 0.244·8-s + 0.333·9-s − 0.0549·10-s + 0.708·11-s − 0.568·12-s − 0.277·13-s + 0.194·14-s + 0.258·15-s + 0.954·16-s + 0.497·17-s − 0.0409·18-s − 1.50·19-s − 0.440·20-s − 0.913·21-s − 0.0870·22-s + 0.123·23-s + 0.140·24-s + 0.200·25-s + 0.0340·26-s + 0.192·27-s + 1.55·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.312099679\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.312099679\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 + 0.173T + 2T^{2} \) |
| 7 | \( 1 + 4.18T + 7T^{2} \) |
| 11 | \( 1 - 2.34T + 11T^{2} \) |
| 17 | \( 1 - 2.05T + 17T^{2} \) |
| 19 | \( 1 + 6.58T + 19T^{2} \) |
| 23 | \( 1 - 0.591T + 23T^{2} \) |
| 29 | \( 1 + 0.846T + 29T^{2} \) |
| 37 | \( 1 - 3.05T + 37T^{2} \) |
| 41 | \( 1 + 1.67T + 41T^{2} \) |
| 43 | \( 1 - 0.550T + 43T^{2} \) |
| 47 | \( 1 + 8.93T + 47T^{2} \) |
| 53 | \( 1 + 4.11T + 53T^{2} \) |
| 59 | \( 1 - 6.30T + 59T^{2} \) |
| 61 | \( 1 + 5.97T + 61T^{2} \) |
| 67 | \( 1 - 2.68T + 67T^{2} \) |
| 71 | \( 1 - 13.9T + 71T^{2} \) |
| 73 | \( 1 - 5.16T + 73T^{2} \) |
| 79 | \( 1 + 2.08T + 79T^{2} \) |
| 83 | \( 1 + 0.818T + 83T^{2} \) |
| 89 | \( 1 + 13.7T + 89T^{2} \) |
| 97 | \( 1 - 10.8T + 97T^{2} \) |
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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
−8.316828715899124132823677688974, −7.40322440669832762779857224210, −6.52757152984901282222602443556, −6.15017550994465039449639850051, −5.14388386302258853206761480827, −4.27653046542049901042297373557, −3.61843531221506725787961060618, −2.94601076251892078327159617013, −1.86688896742135572586512278182, −0.59128121382063963851960314449,
0.59128121382063963851960314449, 1.86688896742135572586512278182, 2.94601076251892078327159617013, 3.61843531221506725787961060618, 4.27653046542049901042297373557, 5.14388386302258853206761480827, 6.15017550994465039449639850051, 6.52757152984901282222602443556, 7.40322440669832762779857224210, 8.316828715899124132823677688974
