L(s) = 1 | − 3-s + 5-s + 2·7-s + 9-s + 11-s − 4·13-s − 15-s + 17-s + 6·19-s − 2·21-s + 2·23-s + 25-s − 27-s − 6·29-s + 4·31-s − 33-s + 2·35-s − 2·37-s + 4·39-s − 6·41-s − 6·43-s + 45-s + 12·47-s − 3·49-s − 51-s + 55-s − 6·57-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s + 0.301·11-s − 1.10·13-s − 0.258·15-s + 0.242·17-s + 1.37·19-s − 0.436·21-s + 0.417·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s + 0.718·31-s − 0.174·33-s + 0.338·35-s − 0.328·37-s + 0.640·39-s − 0.937·41-s − 0.914·43-s + 0.149·45-s + 1.75·47-s − 3/7·49-s − 0.140·51-s + 0.134·55-s − 0.794·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−14.94134850909100, −14.29145507782342, −13.99930745410409, −13.33841184700576, −12.86222881113577, −12.11237945810305, −11.83167999473755, −11.41220951376267, −10.76338821690234, −10.10770638196424, −9.867054417203744, −9.119674596620246, −8.733806294988588, −7.748049113101882, −7.522743500102933, −6.916587060084607, −6.256589611802490, −5.578414195687734, −5.094373294399459, −4.773094991227552, −3.924342736769557, −3.179727891917207, −2.441444510358023, −1.635651840129958, −1.084266185932700, 0,
1.084266185932700, 1.635651840129958, 2.441444510358023, 3.179727891917207, 3.924342736769557, 4.773094991227552, 5.094373294399459, 5.578414195687734, 6.256589611802490, 6.916587060084607, 7.522743500102933, 7.748049113101882, 8.733806294988588, 9.119674596620246, 9.867054417203744, 10.10770638196424, 10.76338821690234, 11.41220951376267, 11.83167999473755, 12.11237945810305, 12.86222881113577, 13.33841184700576, 13.99930745410409, 14.29145507782342, 14.94134850909100