| L(s) = 1 | + 2-s + 3-s + 4-s − 3.07·5-s + 6-s − 1.48·7-s + 8-s + 9-s − 3.07·10-s + 4.90·11-s + 12-s − 3.70·13-s − 1.48·14-s − 3.07·15-s + 16-s − 17-s + 18-s + 0.374·19-s − 3.07·20-s − 1.48·21-s + 4.90·22-s − 3.75·23-s + 24-s + 4.43·25-s − 3.70·26-s + 27-s − 1.48·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.37·5-s + 0.408·6-s − 0.561·7-s + 0.353·8-s + 0.333·9-s − 0.971·10-s + 1.47·11-s + 0.288·12-s − 1.02·13-s − 0.397·14-s − 0.793·15-s + 0.250·16-s − 0.242·17-s + 0.235·18-s + 0.0858·19-s − 0.687·20-s − 0.324·21-s + 1.04·22-s − 0.783·23-s + 0.204·24-s + 0.887·25-s − 0.726·26-s + 0.192·27-s − 0.280·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.805132895\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.805132895\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| good | 5 | \( 1 + 3.07T + 5T^{2} \) |
| 7 | \( 1 + 1.48T + 7T^{2} \) |
| 11 | \( 1 - 4.90T + 11T^{2} \) |
| 13 | \( 1 + 3.70T + 13T^{2} \) |
| 19 | \( 1 - 0.374T + 19T^{2} \) |
| 23 | \( 1 + 3.75T + 23T^{2} \) |
| 29 | \( 1 - 4.03T + 29T^{2} \) |
| 31 | \( 1 - 5.86T + 31T^{2} \) |
| 37 | \( 1 - 8.42T + 37T^{2} \) |
| 41 | \( 1 + 7.48T + 41T^{2} \) |
| 43 | \( 1 - 3.86T + 43T^{2} \) |
| 47 | \( 1 - 5.58T + 47T^{2} \) |
| 53 | \( 1 - 0.592T + 53T^{2} \) |
| 61 | \( 1 + 8.24T + 61T^{2} \) |
| 67 | \( 1 - 2.59T + 67T^{2} \) |
| 71 | \( 1 - 0.437T + 71T^{2} \) |
| 73 | \( 1 - 4.15T + 73T^{2} \) |
| 79 | \( 1 - 4.93T + 79T^{2} \) |
| 83 | \( 1 - 15.7T + 83T^{2} \) |
| 89 | \( 1 - 4.17T + 89T^{2} \) |
| 97 | \( 1 - 15.5T + 97T^{2} \) |
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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
−7.909342659360380587863411136655, −7.40280122293042457826913497283, −6.62020076119816520296324555345, −6.17375027357639602795366525244, −4.82535873313550597992822466975, −4.32997555972557677470741058068, −3.69041055798282554964958492705, −3.06484869787967151585441493226, −2.12523795204134110774183244217, −0.76071360635245539395335503538,
0.76071360635245539395335503538, 2.12523795204134110774183244217, 3.06484869787967151585441493226, 3.69041055798282554964958492705, 4.32997555972557677470741058068, 4.82535873313550597992822466975, 6.17375027357639602795366525244, 6.62020076119816520296324555345, 7.40280122293042457826913497283, 7.909342659360380587863411136655
