L(s) = 1 | − 3-s + 5-s − 7-s + 9-s − 11-s − 6·13-s − 15-s − 7·17-s + 5·19-s + 21-s + 23-s + 25-s − 27-s − 5·29-s + 8·31-s + 33-s − 35-s − 2·37-s + 6·39-s + 12·41-s + 11·43-s + 45-s − 8·47-s + 49-s + 7·51-s − 11·53-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 − 1.66·13-s − 0.258·15-s − 1.69·17-s + 1.14·19-s + 0.218·21-s + 0.208·23-s + 1/5·25-s − 0.192·27-s − 0.928·29-s + 1.43·31-s + 0.174·33-s − 0.169·35-s − 0.328·37-s + 0.960·39-s + 1.87·41-s + 1.67·43-s + 0.149·45-s − 1.16·47-s + 1/7·49-s + 0.980·51-s − 1.51·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
good | 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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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
−16.10008841843176, −15.58104863864827, −15.01881572674131, −14.34011911219085, −13.87607949917729, −13.12903935275993, −12.79389126337932, −12.25434027108349, −11.50866116854455, −11.16380957448562, −10.41684874566037, −9.880267578463368, −9.369800869322174, −8.971362692036588, −7.886753568030484, −7.436745824163245, −6.823926003490168, −6.218325896684952, −5.617682758060527, −4.792299498135356, −4.598021504694343, −3.515713448785239, −2.558221201996797, −2.194758099003395, −0.9386306883423345, 0,
0.9386306883423345, 2.194758099003395, 2.558221201996797, 3.515713448785239, 4.598021504694343, 4.792299498135356, 5.617682758060527, 6.218325896684952, 6.823926003490168, 7.436745824163245, 7.886753568030484, 8.971362692036588, 9.369800869322174, 9.880267578463368, 10.41684874566037, 11.16380957448562, 11.50866116854455, 12.25434027108349, 12.79389126337932, 13.12903935275993, 13.87607949917729, 14.34011911219085, 15.01881572674131, 15.58104863864827, 16.10008841843176