| L(s) = 1 | − 2-s + 3.31·3-s + 4-s + 3.08·5-s − 3.31·6-s − 2.82·7-s − 8-s + 7.98·9-s − 3.08·10-s − 2.61·11-s + 3.31·12-s − 4.12·13-s + 2.82·14-s + 10.2·15-s + 16-s + 2.48·17-s − 7.98·18-s − 0.552·19-s + 3.08·20-s − 9.34·21-s + 2.61·22-s + 6.37·23-s − 3.31·24-s + 4.49·25-s + 4.12·26-s + 16.5·27-s − 2.82·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.91·3-s + 0.5·4-s + 1.37·5-s − 1.35·6-s − 1.06·7-s − 0.353·8-s + 2.66·9-s − 0.974·10-s − 0.789·11-s + 0.956·12-s − 1.14·13-s + 0.753·14-s + 2.63·15-s + 0.250·16-s + 0.603·17-s − 1.88·18-s − 0.126·19-s + 0.688·20-s − 2.03·21-s + 0.558·22-s + 1.32·23-s − 0.676·24-s + 0.898·25-s + 0.809·26-s + 3.17·27-s − 0.533·28-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
| \(L(1)\) |
\(\approx\) |
\(3.344828546\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.344828546\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.31T + 3T^{2} \) |
| 5 | \( 1 - 3.08T + 5T^{2} \) |
| 7 | \( 1 + 2.82T + 7T^{2} \) |
| 11 | \( 1 + 2.61T + 11T^{2} \) |
| 13 | \( 1 + 4.12T + 13T^{2} \) |
| 17 | \( 1 - 2.48T + 17T^{2} \) |
| 19 | \( 1 + 0.552T + 19T^{2} \) |
| 23 | \( 1 - 6.37T + 23T^{2} \) |
| 29 | \( 1 - 6.01T + 29T^{2} \) |
| 31 | \( 1 - 5.34T + 31T^{2} \) |
| 37 | \( 1 + 5.21T + 37T^{2} \) |
| 41 | \( 1 + 0.950T + 41T^{2} \) |
| 43 | \( 1 - 6.48T + 43T^{2} \) |
| 47 | \( 1 - 11.4T + 47T^{2} \) |
| 53 | \( 1 - 7.32T + 53T^{2} \) |
| 59 | \( 1 - 0.0104T + 59T^{2} \) |
| 61 | \( 1 - 1.34T + 61T^{2} \) |
| 67 | \( 1 + 5.44T + 67T^{2} \) |
| 71 | \( 1 - 1.50T + 71T^{2} \) |
| 73 | \( 1 + 4.08T + 73T^{2} \) |
| 79 | \( 1 + 9.36T + 79T^{2} \) |
| 83 | \( 1 - 5.90T + 83T^{2} \) |
| 89 | \( 1 + 13.8T + 89T^{2} \) |
| 97 | \( 1 - 10.0T + 97T^{2} \) |
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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
−8.782374023732376088139690100834, −7.77082196241633338720002047565, −7.19457755286442218276511603092, −6.58632400972953766974206279290, −5.56669902435335597204531866964, −4.58298859241132381200728174255, −3.30046961785444449827079360210, −2.65344395823362222586773130873, −2.32542720864985748181597118158, −1.10617843633566767503939276412,
1.10617843633566767503939276412, 2.32542720864985748181597118158, 2.65344395823362222586773130873, 3.30046961785444449827079360210, 4.58298859241132381200728174255, 5.56669902435335597204531866964, 6.58632400972953766974206279290, 7.19457755286442218276511603092, 7.77082196241633338720002047565, 8.782374023732376088139690100834
