| L(s) = 1 | − 2-s + 1.08·3-s + 4-s + 5-s − 1.08·6-s + 3.52·7-s − 8-s − 1.82·9-s − 10-s + 3.17·11-s + 1.08·12-s − 0.154·13-s − 3.52·14-s + 1.08·15-s + 16-s + 4.06·17-s + 1.82·18-s + 6.33·19-s + 20-s + 3.81·21-s − 3.17·22-s − 8.52·23-s − 1.08·24-s + 25-s + 0.154·26-s − 5.22·27-s + 3.52·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.625·3-s + 0.5·4-s + 0.447·5-s − 0.442·6-s + 1.33·7-s − 0.353·8-s − 0.609·9-s − 0.316·10-s + 0.956·11-s + 0.312·12-s − 0.0429·13-s − 0.940·14-s + 0.279·15-s + 0.250·16-s + 0.985·17-s + 0.430·18-s + 1.45·19-s + 0.223·20-s + 0.831·21-s − 0.676·22-s − 1.77·23-s − 0.221·24-s + 0.200·25-s + 0.0303·26-s − 1.00·27-s + 0.665·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.600631688\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.600631688\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 601 | \( 1 + T \) |
| good | 3 | \( 1 - 1.08T + 3T^{2} \) |
| 7 | \( 1 - 3.52T + 7T^{2} \) |
| 11 | \( 1 - 3.17T + 11T^{2} \) |
| 13 | \( 1 + 0.154T + 13T^{2} \) |
| 17 | \( 1 - 4.06T + 17T^{2} \) |
| 19 | \( 1 - 6.33T + 19T^{2} \) |
| 23 | \( 1 + 8.52T + 23T^{2} \) |
| 29 | \( 1 - 8.41T + 29T^{2} \) |
| 31 | \( 1 - 2.25T + 31T^{2} \) |
| 37 | \( 1 + 6.29T + 37T^{2} \) |
| 41 | \( 1 + 3.58T + 41T^{2} \) |
| 43 | \( 1 - 5.40T + 43T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 - 0.747T + 53T^{2} \) |
| 59 | \( 1 - 6.52T + 59T^{2} \) |
| 61 | \( 1 - 1.99T + 61T^{2} \) |
| 67 | \( 1 + 9.55T + 67T^{2} \) |
| 71 | \( 1 + 4.63T + 71T^{2} \) |
| 73 | \( 1 + 4.36T + 73T^{2} \) |
| 79 | \( 1 + 7.48T + 79T^{2} \) |
| 83 | \( 1 - 7.26T + 83T^{2} \) |
| 89 | \( 1 - 10.5T + 89T^{2} \) |
| 97 | \( 1 + 0.695T + 97T^{2} \) |
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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
−8.083039257262280787292552651573, −7.70343085329038770403432295410, −6.86188848229869173521716590133, −5.87650150264447549037981649080, −5.42485461622202111496258310828, −4.37980375727870141011584005710, −3.46282960399219398185259285012, −2.59963709363915295024157457380, −1.73391634097421644177620246239, −0.991880433998681077464306377184,
0.991880433998681077464306377184, 1.73391634097421644177620246239, 2.59963709363915295024157457380, 3.46282960399219398185259285012, 4.37980375727870141011584005710, 5.42485461622202111496258310828, 5.87650150264447549037981649080, 6.86188848229869173521716590133, 7.70343085329038770403432295410, 8.083039257262280787292552651573
