| L(s) = 1 | + 2.54·2-s − 1.32·3-s + 4.48·4-s − 2.35·5-s − 3.38·6-s + 6.31·8-s − 1.23·9-s − 6.00·10-s − 0.753·11-s − 5.95·12-s − 2.77·13-s + 3.13·15-s + 7.12·16-s − 0.102·17-s − 3.13·18-s + 0.742·19-s − 10.5·20-s − 1.91·22-s − 2.97·23-s − 8.40·24-s + 0.561·25-s − 7.06·26-s + 5.62·27-s − 9.61·29-s + 7.98·30-s + 0.217·31-s + 5.49·32-s + ⋯ |
| L(s) = 1 | + 1.80·2-s − 0.767·3-s + 2.24·4-s − 1.05·5-s − 1.38·6-s + 2.23·8-s − 0.410·9-s − 1.89·10-s − 0.227·11-s − 1.72·12-s − 0.769·13-s + 0.809·15-s + 1.78·16-s − 0.0249·17-s − 0.739·18-s + 0.170·19-s − 2.36·20-s − 0.408·22-s − 0.619·23-s − 1.71·24-s + 0.112·25-s − 1.38·26-s + 1.08·27-s − 1.78·29-s + 1.45·30-s + 0.0390·31-s + 0.971·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 - 2.54T + 2T^{2} \) |
| 3 | \( 1 + 1.32T + 3T^{2} \) |
| 5 | \( 1 + 2.35T + 5T^{2} \) |
| 11 | \( 1 + 0.753T + 11T^{2} \) |
| 13 | \( 1 + 2.77T + 13T^{2} \) |
| 17 | \( 1 + 0.102T + 17T^{2} \) |
| 19 | \( 1 - 0.742T + 19T^{2} \) |
| 23 | \( 1 + 2.97T + 23T^{2} \) |
| 29 | \( 1 + 9.61T + 29T^{2} \) |
| 31 | \( 1 - 0.217T + 31T^{2} \) |
| 37 | \( 1 + 5.11T + 37T^{2} \) |
| 43 | \( 1 + 2.13T + 43T^{2} \) |
| 47 | \( 1 + 10.3T + 47T^{2} \) |
| 53 | \( 1 - 3.20T + 53T^{2} \) |
| 59 | \( 1 - 8.35T + 59T^{2} \) |
| 61 | \( 1 + 7.56T + 61T^{2} \) |
| 67 | \( 1 - 15.5T + 67T^{2} \) |
| 71 | \( 1 - 9.88T + 71T^{2} \) |
| 73 | \( 1 + 15.4T + 73T^{2} \) |
| 79 | \( 1 + 13.8T + 79T^{2} \) |
| 83 | \( 1 - 3.89T + 83T^{2} \) |
| 89 | \( 1 - 6.30T + 89T^{2} \) |
| 97 | \( 1 - 8.28T + 97T^{2} \) |
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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
−8.486980311247604872866774851885, −7.57631785577464024275462998845, −6.98156561903986120272588374780, −6.08305315433953894611637173277, −5.37352499917259919759686491979, −4.79406694484375284756581263195, −3.88812083502803785544539319516, −3.21013045129696189020198780627, −2.06936943509835074608922729464, 0,
2.06936943509835074608922729464, 3.21013045129696189020198780627, 3.88812083502803785544539319516, 4.79406694484375284756581263195, 5.37352499917259919759686491979, 6.08305315433953894611637173277, 6.98156561903986120272588374780, 7.57631785577464024275462998845, 8.486980311247604872866774851885
