| L(s) = 1 | − 2.77·2-s − 0.417·3-s + 5.70·4-s + 0.239·5-s + 1.15·6-s + 2.35·7-s − 10.2·8-s − 2.82·9-s − 0.665·10-s + 5.59·11-s − 2.38·12-s + 1.16·13-s − 6.52·14-s − 0.100·15-s + 17.1·16-s + 17-s + 7.84·18-s − 5.90·19-s + 1.36·20-s − 0.981·21-s − 15.5·22-s + 6.51·23-s + 4.29·24-s − 4.94·25-s − 3.22·26-s + 2.43·27-s + 13.4·28-s + ⋯ |
| L(s) = 1 | − 1.96·2-s − 0.241·3-s + 2.85·4-s + 0.107·5-s + 0.473·6-s + 0.888·7-s − 3.63·8-s − 0.941·9-s − 0.210·10-s + 1.68·11-s − 0.687·12-s + 0.321·13-s − 1.74·14-s − 0.0258·15-s + 4.28·16-s + 0.242·17-s + 1.84·18-s − 1.35·19-s + 0.305·20-s − 0.214·21-s − 3.30·22-s + 1.35·23-s + 0.875·24-s − 0.988·25-s − 0.631·26-s + 0.468·27-s + 2.53·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8824652655\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8824652655\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 - T \) |
| 353 | \( 1 + T \) |
| good | 2 | \( 1 + 2.77T + 2T^{2} \) |
| 3 | \( 1 + 0.417T + 3T^{2} \) |
| 5 | \( 1 - 0.239T + 5T^{2} \) |
| 7 | \( 1 - 2.35T + 7T^{2} \) |
| 11 | \( 1 - 5.59T + 11T^{2} \) |
| 13 | \( 1 - 1.16T + 13T^{2} \) |
| 19 | \( 1 + 5.90T + 19T^{2} \) |
| 23 | \( 1 - 6.51T + 23T^{2} \) |
| 29 | \( 1 + 0.502T + 29T^{2} \) |
| 31 | \( 1 - 7.71T + 31T^{2} \) |
| 37 | \( 1 - 4.01T + 37T^{2} \) |
| 41 | \( 1 - 6.35T + 41T^{2} \) |
| 43 | \( 1 + 1.67T + 43T^{2} \) |
| 47 | \( 1 + 4.93T + 47T^{2} \) |
| 53 | \( 1 - 4.29T + 53T^{2} \) |
| 59 | \( 1 - 1.89T + 59T^{2} \) |
| 61 | \( 1 - 0.183T + 61T^{2} \) |
| 67 | \( 1 + 12.3T + 67T^{2} \) |
| 71 | \( 1 - 15.2T + 71T^{2} \) |
| 73 | \( 1 + 9.63T + 73T^{2} \) |
| 79 | \( 1 + 5.10T + 79T^{2} \) |
| 83 | \( 1 + 1.18T + 83T^{2} \) |
| 89 | \( 1 - 11.5T + 89T^{2} \) |
| 97 | \( 1 - 15.1T + 97T^{2} \) |
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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
−8.199609001927542879806888549902, −7.70594369444200167691885353065, −6.66411076123144163721512518451, −6.36782389686239214957460143027, −5.63812491694383977101394092200, −4.41432881492290783983086882573, −3.28327068756268751053627560331, −2.33216650939697563707474801082, −1.49643251871021164438443349673, −0.72031892529592850345975814336,
0.72031892529592850345975814336, 1.49643251871021164438443349673, 2.33216650939697563707474801082, 3.28327068756268751053627560331, 4.41432881492290783983086882573, 5.63812491694383977101394092200, 6.36782389686239214957460143027, 6.66411076123144163721512518451, 7.70594369444200167691885353065, 8.199609001927542879806888549902
