| L(s) = 1 | + 2.21·2-s + 3.13·3-s + 2.90·4-s − 4.00·5-s + 6.94·6-s + 1.99·8-s + 6.85·9-s − 8.87·10-s + 11-s + 9.11·12-s + 13-s − 12.5·15-s − 1.38·16-s + 5.99·17-s + 15.1·18-s − 3.39·19-s − 11.6·20-s + 2.21·22-s + 0.438·23-s + 6.27·24-s + 11.0·25-s + 2.21·26-s + 12.0·27-s + 5.37·29-s − 27.8·30-s + 1.24·31-s − 7.05·32-s + ⋯ |
| L(s) = 1 | + 1.56·2-s + 1.81·3-s + 1.45·4-s − 1.79·5-s + 2.83·6-s + 0.706·8-s + 2.28·9-s − 2.80·10-s + 0.301·11-s + 2.62·12-s + 0.277·13-s − 3.24·15-s − 0.345·16-s + 1.45·17-s + 3.57·18-s − 0.778·19-s − 2.60·20-s + 0.472·22-s + 0.0913·23-s + 1.28·24-s + 2.21·25-s + 0.434·26-s + 2.32·27-s + 0.998·29-s − 5.08·30-s + 0.224·31-s − 1.24·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.934012574\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.934012574\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - 2.21T + 2T^{2} \) |
| 3 | \( 1 - 3.13T + 3T^{2} \) |
| 5 | \( 1 + 4.00T + 5T^{2} \) |
| 17 | \( 1 - 5.99T + 17T^{2} \) |
| 19 | \( 1 + 3.39T + 19T^{2} \) |
| 23 | \( 1 - 0.438T + 23T^{2} \) |
| 29 | \( 1 - 5.37T + 29T^{2} \) |
| 31 | \( 1 - 1.24T + 31T^{2} \) |
| 37 | \( 1 - 2.61T + 37T^{2} \) |
| 41 | \( 1 - 0.391T + 41T^{2} \) |
| 43 | \( 1 - 12.5T + 43T^{2} \) |
| 47 | \( 1 - 2.70T + 47T^{2} \) |
| 53 | \( 1 - 13.6T + 53T^{2} \) |
| 59 | \( 1 - 8.80T + 59T^{2} \) |
| 61 | \( 1 + 10.6T + 61T^{2} \) |
| 67 | \( 1 - 7.29T + 67T^{2} \) |
| 71 | \( 1 + 11.8T + 71T^{2} \) |
| 73 | \( 1 + 7.84T + 73T^{2} \) |
| 79 | \( 1 + 6.49T + 79T^{2} \) |
| 83 | \( 1 - 0.250T + 83T^{2} \) |
| 89 | \( 1 + 5.99T + 89T^{2} \) |
| 97 | \( 1 + 4.60T + 97T^{2} \) |
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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
−7.73367013717120472811629611111, −7.40091128107551122742823223703, −6.68836945028450202254378039031, −5.68999235779297196565881023456, −4.56975650739327668688413804255, −4.12601861908262635458645294362, −3.70997257677000196745902718798, −3.01128632965121567556976007358, −2.50754468936602960543867694527, −1.09606987013489424984925515295,
1.09606987013489424984925515295, 2.50754468936602960543867694527, 3.01128632965121567556976007358, 3.70997257677000196745902718798, 4.12601861908262635458645294362, 4.56975650739327668688413804255, 5.68999235779297196565881023456, 6.68836945028450202254378039031, 7.40091128107551122742823223703, 7.73367013717120472811629611111
