| L(s) = 1 | − 4-s + 5·7-s − 6·11-s + 16-s − 2·25-s − 5·28-s − 7·37-s − 3·41-s + 6·44-s + 12·47-s + 7·49-s − 12·53-s − 64-s − 7·67-s − 9·71-s − 13·73-s − 30·77-s − 6·83-s + 2·100-s + 15·101-s − 24·107-s + 5·112-s + 14·121-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1.88·7-s − 1.80·11-s + 1/4·16-s − 2/5·25-s − 0.944·28-s − 1.15·37-s − 0.468·41-s + 0.904·44-s + 1.75·47-s + 49-s − 1.64·53-s − 1/8·64-s − 0.855·67-s − 1.06·71-s − 1.52·73-s − 3.41·77-s − 0.658·83-s + 1/5·100-s + 1.49·101-s − 2.32·107-s + 0.472·112-s + 1.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 443556 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 443556 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | | \( 1 \) |
| 37 | $C_2$ | \( 1 + 7 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 43 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 49 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
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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
−8.395578161648860251431918840109, −7.897352421254262823669130339633, −7.53097257797602820565882923688, −7.32234291545755653125329158509, −6.48680973333868110491697318176, −5.77651480907957870188964312676, −5.43718582605932237480785343826, −5.02892763163288046744298888993, −4.57776032053610990715632743670, −4.21047289704946284464629968177, −3.32312081745594526576384257059, −2.72798655451542226626066913615, −1.97940733659948740649294559830, −1.38007848639479498277233592714, 0,
1.38007848639479498277233592714, 1.97940733659948740649294559830, 2.72798655451542226626066913615, 3.32312081745594526576384257059, 4.21047289704946284464629968177, 4.57776032053610990715632743670, 5.02892763163288046744298888993, 5.43718582605932237480785343826, 5.77651480907957870188964312676, 6.48680973333868110491697318176, 7.32234291545755653125329158509, 7.53097257797602820565882923688, 7.897352421254262823669130339633, 8.395578161648860251431918840109
