| L(s) = 1 | − 2-s + 1.08·3-s + 4-s + 1.70·5-s − 1.08·6-s − 3.98·7-s − 8-s − 1.82·9-s − 1.70·10-s − 3.98·11-s + 1.08·12-s − 0.477·13-s + 3.98·14-s + 1.84·15-s + 16-s + 0.273·17-s + 1.82·18-s − 3.88·19-s + 1.70·20-s − 4.32·21-s + 3.98·22-s − 23-s − 1.08·24-s − 2.10·25-s + 0.477·26-s − 5.23·27-s − 3.98·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.625·3-s + 0.5·4-s + 0.761·5-s − 0.442·6-s − 1.50·7-s − 0.353·8-s − 0.608·9-s − 0.538·10-s − 1.20·11-s + 0.312·12-s − 0.132·13-s + 1.06·14-s + 0.476·15-s + 0.250·16-s + 0.0662·17-s + 0.430·18-s − 0.891·19-s + 0.380·20-s − 0.943·21-s + 0.849·22-s − 0.208·23-s − 0.221·24-s − 0.420·25-s + 0.0936·26-s − 1.00·27-s − 0.753·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9319426850\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9319426850\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 - T \) |
| good | 3 | \( 1 - 1.08T + 3T^{2} \) |
| 5 | \( 1 - 1.70T + 5T^{2} \) |
| 7 | \( 1 + 3.98T + 7T^{2} \) |
| 11 | \( 1 + 3.98T + 11T^{2} \) |
| 13 | \( 1 + 0.477T + 13T^{2} \) |
| 17 | \( 1 - 0.273T + 17T^{2} \) |
| 19 | \( 1 + 3.88T + 19T^{2} \) |
| 29 | \( 1 + 6.49T + 29T^{2} \) |
| 31 | \( 1 - 9.66T + 31T^{2} \) |
| 37 | \( 1 - 2.31T + 37T^{2} \) |
| 41 | \( 1 - 7.13T + 41T^{2} \) |
| 43 | \( 1 - 8.64T + 43T^{2} \) |
| 47 | \( 1 - 6.68T + 47T^{2} \) |
| 53 | \( 1 - 0.222T + 53T^{2} \) |
| 59 | \( 1 + 12.6T + 59T^{2} \) |
| 61 | \( 1 + 2.37T + 61T^{2} \) |
| 67 | \( 1 - 3.22T + 67T^{2} \) |
| 71 | \( 1 - 4.01T + 71T^{2} \) |
| 73 | \( 1 + 3.72T + 73T^{2} \) |
| 79 | \( 1 - 15.2T + 79T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 + 4.75T + 89T^{2} \) |
| 97 | \( 1 - 1.58T + 97T^{2} \) |
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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
−8.034512303935145407232054060229, −7.63782937213581864003739102458, −6.64432713605409264115785086209, −5.97716745109208322703214957397, −5.62046292763210893252213872059, −4.29446043529679817550622885892, −3.27349122789306474741618748622, −2.59590738601934520443321517739, −2.14988679092537557830292116480, −0.50372494134555622034831490648,
0.50372494134555622034831490648, 2.14988679092537557830292116480, 2.59590738601934520443321517739, 3.27349122789306474741618748622, 4.29446043529679817550622885892, 5.62046292763210893252213872059, 5.97716745109208322703214957397, 6.64432713605409264115785086209, 7.63782937213581864003739102458, 8.034512303935145407232054060229
