L(s) = 1 | + 5-s + 7-s − 11-s − 2·13-s − 6·17-s − 4·19-s + 25-s + 6·29-s + 35-s + 6·37-s + 10·41-s − 4·43-s + 8·47-s + 49-s + 6·53-s − 55-s − 8·59-s + 2·61-s − 2·65-s + 8·67-s − 8·71-s − 14·73-s − 77-s − 4·79-s − 16·83-s − 6·85-s + 10·89-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 0.377·7-s − 0.301·11-s − 0.554·13-s − 1.45·17-s − 0.917·19-s + 1/5·25-s + 1.11·29-s + 0.169·35-s + 0.986·37-s + 1.56·41-s − 0.609·43-s + 1.16·47-s + 1/7·49-s + 0.824·53-s − 0.134·55-s − 1.04·59-s + 0.256·61-s − 0.248·65-s + 0.977·67-s − 0.949·71-s − 1.63·73-s − 0.113·77-s − 0.450·79-s − 1.75·83-s − 0.650·85-s + 1.05·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.59254050623377, −14.22290687710059, −13.59943635286332, −13.11224370016802, −12.73807750638779, −12.17168831725965, −11.47027566567647, −11.15159896186820, −10.40411709012911, −10.23770401721900, −9.458643665349820, −8.810790555022448, −8.651193166469623, −7.804213156737895, −7.357465323644798, −6.688031783562912, −6.189046858697531, −5.669081112960051, −4.892424920013265, −4.424380717232947, −4.019506981786405, −2.818465055901580, −2.510519406185388, −1.859386425515667, −0.9480985034529845, 0,
0.9480985034529845, 1.859386425515667, 2.510519406185388, 2.818465055901580, 4.019506981786405, 4.424380717232947, 4.892424920013265, 5.669081112960051, 6.189046858697531, 6.688031783562912, 7.357465323644798, 7.804213156737895, 8.651193166469623, 8.810790555022448, 9.458643665349820, 10.23770401721900, 10.40411709012911, 11.15159896186820, 11.47027566567647, 12.17168831725965, 12.73807750638779, 13.11224370016802, 13.59943635286332, 14.22290687710059, 14.59254050623377