| L(s) = 1 | − 5-s − 11-s − 5·13-s − 8·17-s − 5·19-s + 8·23-s + 5·25-s − 3·29-s − 2·31-s + 3·37-s + 6·41-s + 7·43-s + 12·47-s − 11·53-s + 55-s − 5·59-s − 20·61-s + 5·65-s − 7·67-s + 4·71-s + 73-s − 8·79-s − 7·83-s + 8·85-s − 6·89-s + 5·95-s − 25·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.301·11-s − 1.38·13-s − 1.94·17-s − 1.14·19-s + 1.66·23-s + 25-s − 0.557·29-s − 0.359·31-s + 0.493·37-s + 0.937·41-s + 1.06·43-s + 1.75·47-s − 1.51·53-s + 0.134·55-s − 0.650·59-s − 2.56·61-s + 0.620·65-s − 0.855·67-s + 0.474·71-s + 0.117·73-s − 0.900·79-s − 0.768·83-s + 0.867·85-s − 0.635·89-s + 0.512·95-s − 2.53·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49787136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49787136 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + T + 8 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 5 T + 18 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 + 5 T + 30 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 3 T + 46 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 3 T + 62 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 7 T + 84 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 11 T + 122 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 5 T + 110 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 + 7 T + 132 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - T + 132 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 117 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 7 T + 164 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 6 T + 130 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 25 T + 336 T^{2} + 25 p T^{3} + p^{2} T^{4} \) |
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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
−7.63724376452629279997646846898, −7.36447868010732954010391574373, −7.17638395811140182612864436941, −6.81051372762930259694065079828, −6.27596183184069741843660313223, −6.26329866346852993259936709351, −5.50387914382833609014061776968, −5.34996742665071223781701488900, −4.78687111054456079028808124550, −4.48768488650455100103462741734, −4.15507683660874113359597880048, −4.13605463462231984861711202378, −3.09936875335235917136732232050, −2.91440792523149992233732402865, −2.59384201415772275546554883824, −2.15615640660818966884308039621, −1.58857712517828833885757977931, −0.972466327584171584956954989671, 0, 0,
0.972466327584171584956954989671, 1.58857712517828833885757977931, 2.15615640660818966884308039621, 2.59384201415772275546554883824, 2.91440792523149992233732402865, 3.09936875335235917136732232050, 4.13605463462231984861711202378, 4.15507683660874113359597880048, 4.48768488650455100103462741734, 4.78687111054456079028808124550, 5.34996742665071223781701488900, 5.50387914382833609014061776968, 6.26329866346852993259936709351, 6.27596183184069741843660313223, 6.81051372762930259694065079828, 7.17638395811140182612864436941, 7.36447868010732954010391574373, 7.63724376452629279997646846898
