| L(s) = 1 | − 5-s − 4·7-s − 2·11-s + 2·13-s − 6·19-s + 25-s − 10·29-s − 8·31-s + 4·35-s − 4·37-s + 10·41-s + 43-s + 9·49-s − 12·53-s + 2·55-s − 6·59-s − 10·61-s − 2·65-s − 12·67-s + 4·71-s − 8·73-s + 8·77-s + 16·79-s − 12·83-s + 10·89-s − 8·91-s + 6·95-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.51·7-s − 0.603·11-s + 0.554·13-s − 1.37·19-s + 1/5·25-s − 1.85·29-s − 1.43·31-s + 0.676·35-s − 0.657·37-s + 1.56·41-s + 0.152·43-s + 9/7·49-s − 1.64·53-s + 0.269·55-s − 0.781·59-s − 1.28·61-s − 0.248·65-s − 1.46·67-s + 0.474·71-s − 0.936·73-s + 0.911·77-s + 1.80·79-s − 1.31·83-s + 1.05·89-s − 0.838·91-s + 0.615·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−15.59838823999830, −15.22267051427458, −14.66664611630287, −14.03117355465891, −13.25588149381954, −12.97339281198927, −12.61538006320702, −12.08605588287216, −11.16023651168422, −10.82510260695572, −10.43001440388356, −9.572704368790627, −9.145002654369899, −8.790841183608106, −7.761178336262213, −7.577426118901042, −6.766862192879689, −6.169783670735022, −5.821240756443732, −4.990704951397567, −4.138725495295343, −3.673764055084741, −3.093146114608665, −2.342290056505818, −1.483029068902879, 0, 0,
1.483029068902879, 2.342290056505818, 3.093146114608665, 3.673764055084741, 4.138725495295343, 4.990704951397567, 5.821240756443732, 6.169783670735022, 6.766862192879689, 7.577426118901042, 7.761178336262213, 8.790841183608106, 9.145002654369899, 9.572704368790627, 10.43001440388356, 10.82510260695572, 11.16023651168422, 12.08605588287216, 12.61538006320702, 12.97339281198927, 13.25588149381954, 14.03117355465891, 14.66664611630287, 15.22267051427458, 15.59838823999830
