| L(s) = 1 | − 2·2-s + 3·4-s + 2·5-s + 2·7-s − 4·8-s + 2·9-s − 4·10-s − 6·13-s − 4·14-s + 5·16-s − 4·18-s + 6·20-s − 25-s + 12·26-s + 6·28-s + 12·29-s − 6·32-s + 4·35-s + 6·36-s − 12·37-s − 8·40-s + 4·45-s + 3·49-s + 2·50-s − 18·52-s − 8·56-s − 24·58-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s + 0.894·5-s + 0.755·7-s − 1.41·8-s + 2/3·9-s − 1.26·10-s − 1.66·13-s − 1.06·14-s + 5/4·16-s − 0.942·18-s + 1.34·20-s − 1/5·25-s + 2.35·26-s + 1.13·28-s + 2.22·29-s − 1.06·32-s + 0.676·35-s + 36-s − 1.97·37-s − 1.26·40-s + 0.596·45-s + 3/7·49-s + 0.282·50-s − 2.49·52-s − 1.06·56-s − 3.15·58-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
| \(L(1)\) |
\(\approx\) |
\(1.203028369\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.203028369\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 138 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−10.44307336728259573659582000160, −9.780227186349321489265749185014, −9.656183385032763111984555662269, −8.906328361576042812970531244050, −8.896613643678269165286507152436, −8.169596799535049908461874339089, −7.87120752163972454266719684618, −7.44364899286821136035155777133, −7.01209298842118364582033703595, −6.58297354953012058025656953974, −6.27322486486610617256680238768, −5.44302050111893144366225578955, −5.18881185588688753441575843163, −4.63666709255054541358336379834, −4.09479762810034253666213061541, −3.07341589045115935917393453897, −2.69661481308176722433592236973, −1.82979207598179890081399088809, −1.74305596022200685665003616262, −0.65568897347295145940115004059,
0.65568897347295145940115004059, 1.74305596022200685665003616262, 1.82979207598179890081399088809, 2.69661481308176722433592236973, 3.07341589045115935917393453897, 4.09479762810034253666213061541, 4.63666709255054541358336379834, 5.18881185588688753441575843163, 5.44302050111893144366225578955, 6.27322486486610617256680238768, 6.58297354953012058025656953974, 7.01209298842118364582033703595, 7.44364899286821136035155777133, 7.87120752163972454266719684618, 8.169596799535049908461874339089, 8.896613643678269165286507152436, 8.906328361576042812970531244050, 9.656183385032763111984555662269, 9.780227186349321489265749185014, 10.44307336728259573659582000160
