| L(s) = 1 | + 4-s − 2·5-s + 2·7-s + 2·9-s − 4·13-s + 16-s − 2·20-s − 25-s + 2·28-s + 2·29-s − 4·35-s + 2·36-s − 12·37-s − 4·45-s − 2·47-s − 3·49-s − 4·52-s + 4·63-s + 64-s + 8·65-s + 10·67-s + 2·73-s + 14·79-s − 2·80-s − 5·81-s − 10·83-s − 8·91-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 0.894·5-s + 0.755·7-s + 2/3·9-s − 1.10·13-s + 1/4·16-s − 0.447·20-s − 1/5·25-s + 0.377·28-s + 0.371·29-s − 0.676·35-s + 1/3·36-s − 1.97·37-s − 0.596·45-s − 0.291·47-s − 3/7·49-s − 0.554·52-s + 0.503·63-s + 1/8·64-s + 0.992·65-s + 1.22·67-s + 0.234·73-s + 1.57·79-s − 0.223·80-s − 5/9·81-s − 1.09·83-s − 0.838·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 828100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 828100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.784355288\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.784355288\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 80 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) |
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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
−8.143732791004829990426230877356, −7.68440705477272645697000053360, −7.40302475863024745409147094400, −7.02711654203804062044679660272, −6.59138247515122962125890996188, −6.06143852868492765066664855266, −5.33565986397676455187038187339, −4.99396516556135196229742434742, −4.57833762752176928725735103000, −4.03193207023829289197003676250, −3.47341281975146449471267738168, −2.98176894892937075830476176140, −2.05171959894902062722093442468, −1.78418141661713073218984014814, −0.63101440855932359243217600564,
0.63101440855932359243217600564, 1.78418141661713073218984014814, 2.05171959894902062722093442468, 2.98176894892937075830476176140, 3.47341281975146449471267738168, 4.03193207023829289197003676250, 4.57833762752176928725735103000, 4.99396516556135196229742434742, 5.33565986397676455187038187339, 6.06143852868492765066664855266, 6.59138247515122962125890996188, 7.02711654203804062044679660272, 7.40302475863024745409147094400, 7.68440705477272645697000053360, 8.143732791004829990426230877356
