| L(s) = 1 | + 2-s − 4-s − 3·8-s + 6·11-s − 16-s + 6·22-s − 2·23-s + 4·25-s − 8·29-s + 5·32-s − 16·43-s − 6·44-s − 2·46-s + 4·50-s − 8·53-s − 8·58-s + 7·64-s − 20·67-s + 2·71-s − 12·79-s − 16·86-s − 18·88-s + 2·92-s − 4·100-s − 8·106-s + 6·107-s − 8·109-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.06·8-s + 1.80·11-s − 1/4·16-s + 1.27·22-s − 0.417·23-s + 4/5·25-s − 1.48·29-s + 0.883·32-s − 2.43·43-s − 0.904·44-s − 0.294·46-s + 0.565·50-s − 1.09·53-s − 1.05·58-s + 7/8·64-s − 2.44·67-s + 0.237·71-s − 1.35·79-s − 1.72·86-s − 1.91·88-s + 0.208·92-s − 2/5·100-s − 0.777·106-s + 0.580·107-s − 0.766·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 156 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
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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
−8.110244290700951261235418325054, −7.51481485643188820316858075483, −7.07141742808539722513764664794, −6.51198773994269405147540523122, −6.24524373900082075636946024781, −5.79683819446981297483644091898, −5.18496159484992186455362015020, −4.78382366782686739394694361341, −4.25097251576206945983178090130, −3.83911261375137372659619105488, −3.35740605144572352369985534493, −2.90149719559107386221934521898, −1.84845573334673391865245762722, −1.29974486145904636586531470782, 0,
1.29974486145904636586531470782, 1.84845573334673391865245762722, 2.90149719559107386221934521898, 3.35740605144572352369985534493, 3.83911261375137372659619105488, 4.25097251576206945983178090130, 4.78382366782686739394694361341, 5.18496159484992186455362015020, 5.79683819446981297483644091898, 6.24524373900082075636946024781, 6.51198773994269405147540523122, 7.07141742808539722513764664794, 7.51481485643188820316858075483, 8.110244290700951261235418325054
