| L(s) = 1 | + 3-s − 5-s + 7-s + 9-s + 11-s + 2·13-s − 15-s + 2·17-s + 4·19-s + 21-s + 8·23-s + 25-s + 27-s − 6·29-s + 8·31-s + 33-s − 35-s + 2·37-s + 2·39-s + 2·41-s − 4·43-s − 45-s + 8·47-s + 49-s + 2·51-s + 10·53-s − 55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.301·11-s + 0.554·13-s − 0.258·15-s + 0.485·17-s + 0.917·19-s + 0.218·21-s + 1.66·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s + 1.43·31-s + 0.174·33-s − 0.169·35-s + 0.328·37-s + 0.320·39-s + 0.312·41-s − 0.609·43-s − 0.149·45-s + 1.16·47-s + 1/7·49-s + 0.280·51-s + 1.37·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 73920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.555408254\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.555408254\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.11173208807014, −13.51466266423754, −13.25070951025900, −12.64345154343537, −11.85805724531515, −11.80355224074710, −10.96756237359714, −10.74624198724065, −9.928227853411735, −9.475168252733490, −8.987141056484410, −8.438880998702010, −7.995118810033877, −7.475972100157875, −6.912396619215306, −6.510462827334979, −5.440998384640953, −5.339523513185030, −4.446530233998434, −3.893753828835292, −3.404356173877633, −2.766365913161134, −2.123945061548687, −1.096138084384254, −0.8401011274025492,
0.8401011274025492, 1.096138084384254, 2.123945061548687, 2.766365913161134, 3.404356173877633, 3.893753828835292, 4.446530233998434, 5.339523513185030, 5.440998384640953, 6.510462827334979, 6.912396619215306, 7.475972100157875, 7.995118810033877, 8.438880998702010, 8.987141056484410, 9.475168252733490, 9.928227853411735, 10.74624198724065, 10.96756237359714, 11.80355224074710, 11.85805724531515, 12.64345154343537, 13.25070951025900, 13.51466266423754, 14.11173208807014
