| L(s) = 1 | + 1.29·2-s − 1.43·3-s − 6.31·4-s − 20.4·5-s − 1.85·6-s − 18.5·8-s − 24.9·9-s − 26.5·10-s − 22.8·11-s + 9.02·12-s + 44.4·13-s + 29.1·15-s + 26.3·16-s + 13.0·17-s − 32.4·18-s − 5.41·19-s + 128.·20-s − 29.7·22-s − 175.·23-s + 26.6·24-s + 291.·25-s + 57.7·26-s + 74.3·27-s + 165.·29-s + 37.9·30-s + 155.·31-s + 182.·32-s + ⋯ |
| L(s) = 1 | + 0.459·2-s − 0.275·3-s − 0.788·4-s − 1.82·5-s − 0.126·6-s − 0.822·8-s − 0.924·9-s − 0.838·10-s − 0.626·11-s + 0.217·12-s + 0.948·13-s + 0.502·15-s + 0.411·16-s + 0.186·17-s − 0.424·18-s − 0.0653·19-s + 1.44·20-s − 0.288·22-s − 1.58·23-s + 0.226·24-s + 2.33·25-s + 0.435·26-s + 0.529·27-s + 1.06·29-s + 0.230·30-s + 0.901·31-s + 1.01·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2107 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2107 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 43 | \( 1 + 43T \) |
| good | 2 | \( 1 - 1.29T + 8T^{2} \) |
| 3 | \( 1 + 1.43T + 27T^{2} \) |
| 5 | \( 1 + 20.4T + 125T^{2} \) |
| 11 | \( 1 + 22.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 44.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 13.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 5.41T + 6.85e3T^{2} \) |
| 23 | \( 1 + 175.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 165.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 155.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 95.3T + 5.06e4T^{2} \) |
| 41 | \( 1 + 189.T + 6.89e4T^{2} \) |
| 47 | \( 1 - 37.2T + 1.03e5T^{2} \) |
| 53 | \( 1 - 559.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 82.3T + 2.05e5T^{2} \) |
| 61 | \( 1 - 640.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 509.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 792.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 612.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 237.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 418.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 113.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.64e3T + 9.12e5T^{2} \) |
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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
−8.414520549392475271375960883907, −7.85234081114123195282115597473, −6.70849190000752200799421300888, −5.82158894878630688426070710051, −5.02257969470104422237736631221, −4.19052920162380503893378837376, −3.60609564133083975518241830315, −2.78503292991922820856789559445, −0.77596027391687280571100104136, 0,
0.77596027391687280571100104136, 2.78503292991922820856789559445, 3.60609564133083975518241830315, 4.19052920162380503893378837376, 5.02257969470104422237736631221, 5.82158894878630688426070710051, 6.70849190000752200799421300888, 7.85234081114123195282115597473, 8.414520549392475271375960883907
