L(s) = 1 | − 2-s − 0.128·3-s + 4-s + 3.87·5-s + 0.128·6-s − 4.13·7-s − 8-s − 2.98·9-s − 3.87·10-s + 0.767·11-s − 0.128·12-s − 4.27·13-s + 4.13·14-s − 0.496·15-s + 16-s − 3.18·17-s + 2.98·18-s + 5.33·19-s + 3.87·20-s + 0.530·21-s − 0.767·22-s − 0.226·23-s + 0.128·24-s + 10.0·25-s + 4.27·26-s + 0.766·27-s − 4.13·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.0739·3-s + 0.5·4-s + 1.73·5-s + 0.0523·6-s − 1.56·7-s − 0.353·8-s − 0.994·9-s − 1.22·10-s + 0.231·11-s − 0.0369·12-s − 1.18·13-s + 1.10·14-s − 0.128·15-s + 0.250·16-s − 0.772·17-s + 0.703·18-s + 1.22·19-s + 0.866·20-s + 0.115·21-s − 0.163·22-s − 0.0471·23-s + 0.0261·24-s + 2.00·25-s + 0.838·26-s + 0.147·27-s − 0.781·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.143645868\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.143645868\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 2017 | \( 1 - T \) |
good | 3 | \( 1 + 0.128T + 3T^{2} \) |
| 5 | \( 1 - 3.87T + 5T^{2} \) |
| 7 | \( 1 + 4.13T + 7T^{2} \) |
| 11 | \( 1 - 0.767T + 11T^{2} \) |
| 13 | \( 1 + 4.27T + 13T^{2} \) |
| 17 | \( 1 + 3.18T + 17T^{2} \) |
| 19 | \( 1 - 5.33T + 19T^{2} \) |
| 23 | \( 1 + 0.226T + 23T^{2} \) |
| 29 | \( 1 - 0.979T + 29T^{2} \) |
| 31 | \( 1 + 1.08T + 31T^{2} \) |
| 37 | \( 1 + 2.26T + 37T^{2} \) |
| 41 | \( 1 - 3.62T + 41T^{2} \) |
| 43 | \( 1 + 7.91T + 43T^{2} \) |
| 47 | \( 1 - 7.45T + 47T^{2} \) |
| 53 | \( 1 - 9.72T + 53T^{2} \) |
| 59 | \( 1 + 3.40T + 59T^{2} \) |
| 61 | \( 1 + 11.5T + 61T^{2} \) |
| 67 | \( 1 - 9.10T + 67T^{2} \) |
| 71 | \( 1 - 12.7T + 71T^{2} \) |
| 73 | \( 1 - 0.693T + 73T^{2} \) |
| 79 | \( 1 - 15.2T + 79T^{2} \) |
| 83 | \( 1 - 15.1T + 83T^{2} \) |
| 89 | \( 1 - 11.0T + 89T^{2} \) |
| 97 | \( 1 - 1.00T + 97T^{2} \) |
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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
−8.811411515084584532634216495684, −7.67425534378059733971612283481, −6.73467935905905562000978883641, −6.42205766090067310423959778133, −5.62524209536951199905871252542, −5.06187066576645622409514612227, −3.45761948857249659541410677965, −2.67143606473108740340242420416, −2.10771113075467464670789459463, −0.64818029210386076059259531321,
0.64818029210386076059259531321, 2.10771113075467464670789459463, 2.67143606473108740340242420416, 3.45761948857249659541410677965, 5.06187066576645622409514612227, 5.62524209536951199905871252542, 6.42205766090067310423959778133, 6.73467935905905562000978883641, 7.67425534378059733971612283481, 8.811411515084584532634216495684