L(s) = 1 | − 3-s + 1.43·5-s + 7-s + 9-s − 11-s + 3.95·13-s − 1.43·15-s + 4.51·17-s − 3·19-s − 21-s − 23-s − 2.95·25-s − 27-s − 10.4·29-s − 0.344·31-s + 33-s + 1.43·35-s − 7.55·37-s − 3.95·39-s + 5.17·41-s − 7.08·43-s + 1.43·45-s − 7.51·47-s + 49-s − 4.51·51-s − 4.95·53-s − 1.43·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.640·5-s + 0.377·7-s + 0.333·9-s − 0.301·11-s + 1.09·13-s − 0.369·15-s + 1.09·17-s − 0.688·19-s − 0.218·21-s − 0.208·23-s − 0.590·25-s − 0.192·27-s − 1.93·29-s − 0.0618·31-s + 0.174·33-s + 0.241·35-s − 1.24·37-s − 0.632·39-s + 0.808·41-s − 1.08·43-s + 0.213·45-s − 1.09·47-s + 0.142·49-s − 0.632·51-s − 0.679·53-s − 0.193·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 - 1.43T + 5T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 - 3.95T + 13T^{2} \) |
| 17 | \( 1 - 4.51T + 17T^{2} \) |
| 19 | \( 1 + 3T + 19T^{2} \) |
| 29 | \( 1 + 10.4T + 29T^{2} \) |
| 31 | \( 1 + 0.344T + 31T^{2} \) |
| 37 | \( 1 + 7.55T + 37T^{2} \) |
| 41 | \( 1 - 5.17T + 41T^{2} \) |
| 43 | \( 1 + 7.08T + 43T^{2} \) |
| 47 | \( 1 + 7.51T + 47T^{2} \) |
| 53 | \( 1 + 4.95T + 53T^{2} \) |
| 59 | \( 1 + 4.77T + 59T^{2} \) |
| 61 | \( 1 - 4.60T + 61T^{2} \) |
| 67 | \( 1 + 10.4T + 67T^{2} \) |
| 71 | \( 1 + 12.1T + 71T^{2} \) |
| 73 | \( 1 - 0.518T + 73T^{2} \) |
| 79 | \( 1 + 1.38T + 79T^{2} \) |
| 83 | \( 1 - 6.41T + 83T^{2} \) |
| 89 | \( 1 + 2.81T + 89T^{2} \) |
| 97 | \( 1 + 6.51T + 97T^{2} \) |
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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
−7.57264549067397283367986501221, −6.69790652083937389527888364820, −5.92750422687943032737100137585, −5.62176161541274878235013552943, −4.83691529809373017014359657551, −3.91518828919337212855124938530, −3.23170272754564179301997811106, −1.93743275009628523181529597889, −1.42777048160049476412864054243, 0,
1.42777048160049476412864054243, 1.93743275009628523181529597889, 3.23170272754564179301997811106, 3.91518828919337212855124938530, 4.83691529809373017014359657551, 5.62176161541274878235013552943, 5.92750422687943032737100137585, 6.69790652083937389527888364820, 7.57264549067397283367986501221