| L(s) = 1 | − 2-s + 4-s − 8-s + 8·11-s + 16-s − 2·17-s + 8·19-s − 8·22-s − 6·25-s − 32-s + 2·34-s − 8·38-s − 20·41-s + 24·43-s + 8·44-s − 14·49-s + 6·50-s − 24·59-s + 64-s − 24·67-s − 2·68-s + 20·73-s + 8·76-s + 20·82-s − 8·83-s − 24·86-s − 8·88-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s + 2.41·11-s + 1/4·16-s − 0.485·17-s + 1.83·19-s − 1.70·22-s − 6/5·25-s − 0.176·32-s + 0.342·34-s − 1.29·38-s − 3.12·41-s + 3.65·43-s + 1.20·44-s − 2·49-s + 0.848·50-s − 3.12·59-s + 1/8·64-s − 2.93·67-s − 0.242·68-s + 2.34·73-s + 0.917·76-s + 2.20·82-s − 0.878·83-s − 2.58·86-s − 0.852·88-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2996352 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2996352 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( 1 + T \) |
| 3 | | \( 1 \) |
| 17 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−7.41858973180027275469518461292, −6.79533660929922109224513153399, −6.75731369592298410865796616700, −6.05886413503827459123865903257, −5.95829119297760022471245554659, −5.33789404096089741989225038155, −4.66119827871442481374467622682, −4.36134938649640087809102403725, −3.72436468921055482710532071668, −3.36770001720799780980729067088, −2.92323665285260202872977027719, −2.04907435637041739111232790805, −1.43438626728056627432143840048, −1.20581906297715352620829947778, 0,
1.20581906297715352620829947778, 1.43438626728056627432143840048, 2.04907435637041739111232790805, 2.92323665285260202872977027719, 3.36770001720799780980729067088, 3.72436468921055482710532071668, 4.36134938649640087809102403725, 4.66119827871442481374467622682, 5.33789404096089741989225038155, 5.95829119297760022471245554659, 6.05886413503827459123865903257, 6.75731369592298410865796616700, 6.79533660929922109224513153399, 7.41858973180027275469518461292
