L(s) = 1 | + 5-s − 7-s + 11-s − 3·17-s − 19-s + 7·23-s + 25-s − 5·29-s − 6·31-s − 35-s + 8·37-s + 2·41-s − 7·43-s + 2·47-s + 49-s − 5·53-s + 55-s + 7·59-s − 7·61-s − 14·67-s − 2·71-s − 12·73-s − 77-s − 3·83-s − 3·85-s + 89-s − 95-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.377·7-s + 0.301·11-s − 0.727·17-s − 0.229·19-s + 1.45·23-s + 1/5·25-s − 0.928·29-s − 1.07·31-s − 0.169·35-s + 1.31·37-s + 0.312·41-s − 1.06·43-s + 0.291·47-s + 1/7·49-s − 0.686·53-s + 0.134·55-s + 0.911·59-s − 0.896·61-s − 1.71·67-s − 0.237·71-s − 1.40·73-s − 0.113·77-s − 0.329·83-s − 0.325·85-s + 0.105·89-s − 0.102·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13860 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
good | 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−16.56787719314794, −15.93498772429573, −15.22003074335095, −14.76514682371691, −14.35504495464692, −13.36252954194227, −13.19092890170413, −12.71245801731836, −11.88023899497104, −11.25467885913277, −10.83491692115675, −10.13549781148491, −9.460025970455768, −9.010380367930908, −8.551220843632686, −7.461860955220676, −7.181710089818785, −6.284878268510926, −5.925799549152605, −5.036774322691365, −4.450855068243863, −3.590432093357313, −2.892783530658035, −2.068293352275671, −1.214686226481560, 0,
1.214686226481560, 2.068293352275671, 2.892783530658035, 3.590432093357313, 4.450855068243863, 5.036774322691365, 5.925799549152605, 6.284878268510926, 7.181710089818785, 7.461860955220676, 8.551220843632686, 9.010380367930908, 9.460025970455768, 10.13549781148491, 10.83491692115675, 11.25467885913277, 11.88023899497104, 12.71245801731836, 13.19092890170413, 13.36252954194227, 14.35504495464692, 14.76514682371691, 15.22003074335095, 15.93498772429573, 16.56787719314794