| L(s) = 1 | − 1.46·2-s − 3-s + 0.148·4-s + 0.335·5-s + 1.46·6-s − 7-s + 2.71·8-s + 9-s − 0.491·10-s + 1.33·11-s − 0.148·12-s + 1.84·13-s + 1.46·14-s − 0.335·15-s − 4.27·16-s − 0.996·17-s − 1.46·18-s + 3.33·19-s + 0.0496·20-s + 21-s − 1.96·22-s + 4.13·23-s − 2.71·24-s − 4.88·25-s − 2.70·26-s − 27-s − 0.148·28-s + ⋯ |
| L(s) = 1 | − 1.03·2-s − 0.577·3-s + 0.0741·4-s + 0.149·5-s + 0.598·6-s − 0.377·7-s + 0.959·8-s + 0.333·9-s − 0.155·10-s + 0.403·11-s − 0.0427·12-s + 0.511·13-s + 0.391·14-s − 0.0865·15-s − 1.06·16-s − 0.241·17-s − 0.345·18-s + 0.764·19-s + 0.0111·20-s + 0.218·21-s − 0.418·22-s + 0.861·23-s − 0.554·24-s − 0.977·25-s − 0.530·26-s − 0.192·27-s − 0.0280·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8755052328\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8755052328\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 - T \) |
| good | 2 | \( 1 + 1.46T + 2T^{2} \) |
| 5 | \( 1 - 0.335T + 5T^{2} \) |
| 11 | \( 1 - 1.33T + 11T^{2} \) |
| 13 | \( 1 - 1.84T + 13T^{2} \) |
| 17 | \( 1 + 0.996T + 17T^{2} \) |
| 19 | \( 1 - 3.33T + 19T^{2} \) |
| 23 | \( 1 - 4.13T + 23T^{2} \) |
| 29 | \( 1 - 6.10T + 29T^{2} \) |
| 31 | \( 1 + 6.72T + 31T^{2} \) |
| 37 | \( 1 - 6.92T + 37T^{2} \) |
| 41 | \( 1 - 9.98T + 41T^{2} \) |
| 43 | \( 1 - 4.08T + 43T^{2} \) |
| 47 | \( 1 + 4.33T + 47T^{2} \) |
| 53 | \( 1 - 5.52T + 53T^{2} \) |
| 59 | \( 1 + 5.94T + 59T^{2} \) |
| 61 | \( 1 - 5.96T + 61T^{2} \) |
| 67 | \( 1 + 13.4T + 67T^{2} \) |
| 71 | \( 1 - 6.45T + 71T^{2} \) |
| 73 | \( 1 - 4.22T + 73T^{2} \) |
| 79 | \( 1 - 4.35T + 79T^{2} \) |
| 83 | \( 1 - 2.11T + 83T^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 - 17.0T + 97T^{2} \) |
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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
−7.72480793338268075413230105277, −7.37854546442358716523482819370, −6.48570957570932467398683923277, −5.92632569275293580741327330887, −5.07400199061024441292836668559, −4.32234514042624191283292798672, −3.57544682498764531920313455053, −2.42581516517794999912776445584, −1.34163002294775142309133513892, −0.63518824195027187403518706524,
0.63518824195027187403518706524, 1.34163002294775142309133513892, 2.42581516517794999912776445584, 3.57544682498764531920313455053, 4.32234514042624191283292798672, 5.07400199061024441292836668559, 5.92632569275293580741327330887, 6.48570957570932467398683923277, 7.37854546442358716523482819370, 7.72480793338268075413230105277
