| L(s) = 1 | + 12·5-s − 24·7-s − 22·11-s − 8·13-s − 28·17-s − 68·19-s − 268·23-s − 18·25-s − 260·29-s + 112·31-s − 288·35-s + 316·37-s − 124·41-s − 148·43-s − 572·47-s − 130·49-s − 76·53-s − 264·55-s − 864·59-s − 136·61-s − 96·65-s − 512·67-s − 724·71-s + 972·73-s + 528·77-s − 528·79-s − 2.11e3·83-s + ⋯ |
| L(s) = 1 | + 1.07·5-s − 1.29·7-s − 0.603·11-s − 0.170·13-s − 0.399·17-s − 0.821·19-s − 2.42·23-s − 0.143·25-s − 1.66·29-s + 0.648·31-s − 1.39·35-s + 1.40·37-s − 0.472·41-s − 0.524·43-s − 1.77·47-s − 0.379·49-s − 0.196·53-s − 0.647·55-s − 1.90·59-s − 0.285·61-s − 0.183·65-s − 0.933·67-s − 1.21·71-s + 1.55·73-s + 0.781·77-s − 0.751·79-s − 2.79·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 156816 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156816 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 12 T + 162 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 24 T + 706 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 8 T + 1310 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 28 T + 5558 T^{2} + 28 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 68 T + 14378 T^{2} + 68 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 268 T + 36214 T^{2} + 268 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 260 T + 63694 T^{2} + 260 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 112 T + 38414 T^{2} - 112 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 316 T + 66254 T^{2} - 316 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 124 T + 133750 T^{2} + 124 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 74 T + p^{3} T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 572 T + 274438 T^{2} + 572 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 76 T + 48098 T^{2} + 76 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 864 T + 592918 T^{2} + 864 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 136 T + 136062 T^{2} + 136 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 512 T + 468662 T^{2} + 512 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 724 T + 694966 T^{2} + 724 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 972 T + 996374 T^{2} - 972 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 528 T + 1052674 T^{2} + 528 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 2112 T + 2198694 T^{2} + 2112 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 288 T - 124782 T^{2} + 288 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 500 T + 1245030 T^{2} - 500 p^{3} T^{3} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.40529560901769179360940352977, −10.05783308804332371999961816570, −9.691943699057387933092912451342, −9.602780152248101710406336583547, −8.904757943762688612686988469143, −8.305205710953796645705507284429, −7.87811740800547413327798666806, −7.41064726322153901475281195952, −6.50910591695764302787481839212, −6.42913890955334638661628131567, −5.76781803290502380869367139818, −5.66994616577643587810566789130, −4.56896407020150818853070963648, −4.28351314502518495303545676599, −3.36512440568639058159585179703, −2.91448017618907003083281129995, −2.01237317689166792041705731153, −1.74515037331068871195572042549, 0, 0,
1.74515037331068871195572042549, 2.01237317689166792041705731153, 2.91448017618907003083281129995, 3.36512440568639058159585179703, 4.28351314502518495303545676599, 4.56896407020150818853070963648, 5.66994616577643587810566789130, 5.76781803290502380869367139818, 6.42913890955334638661628131567, 6.50910591695764302787481839212, 7.41064726322153901475281195952, 7.87811740800547413327798666806, 8.305205710953796645705507284429, 8.904757943762688612686988469143, 9.602780152248101710406336583547, 9.691943699057387933092912451342, 10.05783308804332371999961816570, 10.40529560901769179360940352977
