| L(s) = 1 | − 2·3-s − 4·5-s + 9-s − 4·11-s + 8·15-s − 4·16-s + 3·25-s − 2·27-s + 8·33-s + 14·37-s − 4·45-s − 47-s + 8·48-s + 2·49-s + 8·53-s + 16·55-s − 14·59-s + 4·67-s − 14·71-s − 6·75-s + 16·80-s + 4·81-s − 6·89-s − 30·97-s − 4·99-s + 8·103-s − 28·111-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.78·5-s + 1/3·9-s − 1.20·11-s + 2.06·15-s − 16-s + 3/5·25-s − 0.384·27-s + 1.39·33-s + 2.30·37-s − 0.596·45-s − 0.145·47-s + 1.15·48-s + 2/7·49-s + 1.09·53-s + 2.15·55-s − 1.82·59-s + 0.488·67-s − 1.66·71-s − 0.692·75-s + 1.78·80-s + 4/9·81-s − 0.635·89-s − 3.04·97-s − 0.402·99-s + 0.788·103-s − 2.65·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5687 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5687 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 3 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 146 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 13 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
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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
−11.65804728130724256694025446198, −11.30029171433773194355604760389, −11.01646951872179987289164413109, −10.29908871913320616466367826254, −9.560450105121024166278128189696, −8.756712303220108720583947491858, −7.88080419030482785528533709129, −7.73342277568481293628938717015, −6.95445208530895871107756968839, −6.07961547832795668017521451632, −5.43270121417121251180042981930, −4.53379052923454063453493313322, −4.02881143809742540553922655552, −2.75333936438503810243981124508, 0,
2.75333936438503810243981124508, 4.02881143809742540553922655552, 4.53379052923454063453493313322, 5.43270121417121251180042981930, 6.07961547832795668017521451632, 6.95445208530895871107756968839, 7.73342277568481293628938717015, 7.88080419030482785528533709129, 8.756712303220108720583947491858, 9.560450105121024166278128189696, 10.29908871913320616466367826254, 11.01646951872179987289164413109, 11.30029171433773194355604760389, 11.65804728130724256694025446198
