| L(s) = 1 | + 1.52·2-s − 3-s + 0.315·4-s − 3.94·5-s − 1.52·6-s + 2.18·7-s − 2.56·8-s + 9-s − 6.00·10-s − 2.84·11-s − 0.315·12-s − 5.78·13-s + 3.32·14-s + 3.94·15-s − 4.53·16-s − 17-s + 1.52·18-s − 1.37·19-s − 1.24·20-s − 2.18·21-s − 4.32·22-s − 2.73·23-s + 2.56·24-s + 10.5·25-s − 8.80·26-s − 27-s + 0.689·28-s + ⋯ |
| L(s) = 1 | + 1.07·2-s − 0.577·3-s + 0.157·4-s − 1.76·5-s − 0.621·6-s + 0.827·7-s − 0.906·8-s + 0.333·9-s − 1.89·10-s − 0.857·11-s − 0.0910·12-s − 1.60·13-s + 0.889·14-s + 1.01·15-s − 1.13·16-s − 0.242·17-s + 0.358·18-s − 0.314·19-s − 0.278·20-s − 0.477·21-s − 0.922·22-s − 0.570·23-s + 0.523·24-s + 2.11·25-s − 1.72·26-s − 0.192·27-s + 0.130·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1913516454\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1913516454\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 157 | \( 1 - T \) |
| good | 2 | \( 1 - 1.52T + 2T^{2} \) |
| 5 | \( 1 + 3.94T + 5T^{2} \) |
| 7 | \( 1 - 2.18T + 7T^{2} \) |
| 11 | \( 1 + 2.84T + 11T^{2} \) |
| 13 | \( 1 + 5.78T + 13T^{2} \) |
| 19 | \( 1 + 1.37T + 19T^{2} \) |
| 23 | \( 1 + 2.73T + 23T^{2} \) |
| 29 | \( 1 + 6.66T + 29T^{2} \) |
| 31 | \( 1 + 1.44T + 31T^{2} \) |
| 37 | \( 1 + 7.60T + 37T^{2} \) |
| 41 | \( 1 - 1.52T + 41T^{2} \) |
| 43 | \( 1 - 4.78T + 43T^{2} \) |
| 47 | \( 1 + 11.0T + 47T^{2} \) |
| 53 | \( 1 + 11.0T + 53T^{2} \) |
| 59 | \( 1 + 9.75T + 59T^{2} \) |
| 61 | \( 1 + 0.151T + 61T^{2} \) |
| 67 | \( 1 - 4.17T + 67T^{2} \) |
| 71 | \( 1 + 4.38T + 71T^{2} \) |
| 73 | \( 1 - 2.70T + 73T^{2} \) |
| 79 | \( 1 - 8.63T + 79T^{2} \) |
| 83 | \( 1 + 6.19T + 83T^{2} \) |
| 89 | \( 1 - 9.61T + 89T^{2} \) |
| 97 | \( 1 - 15.3T + 97T^{2} \) |
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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
−7.74568807590258546239981575421, −7.19694122542032770887752998546, −6.36368925318699569942887391001, −5.35365086155485373742462437204, −4.87471462540443452880751734704, −4.50904382056488040851276982530, −3.74060445231281656363420876337, −3.01234697778015261268270061314, −1.96693085430609144510262647811, −0.17777340622324070137404943380,
0.17777340622324070137404943380, 1.96693085430609144510262647811, 3.01234697778015261268270061314, 3.74060445231281656363420876337, 4.50904382056488040851276982530, 4.87471462540443452880751734704, 5.35365086155485373742462437204, 6.36368925318699569942887391001, 7.19694122542032770887752998546, 7.74568807590258546239981575421
