| L(s) = 1 | − 2·2-s + 4-s + 2·5-s + 4·7-s + 2·9-s − 4·10-s − 2·11-s − 8·14-s + 16-s + 8·17-s − 4·18-s + 2·20-s + 4·22-s + 3·25-s + 4·28-s + 4·29-s + 2·32-s − 16·34-s + 8·35-s + 2·36-s + 4·37-s − 12·41-s − 12·43-s − 2·44-s + 4·45-s − 2·49-s − 6·50-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1/2·4-s + 0.894·5-s + 1.51·7-s + 2/3·9-s − 1.26·10-s − 0.603·11-s − 2.13·14-s + 1/4·16-s + 1.94·17-s − 0.942·18-s + 0.447·20-s + 0.852·22-s + 3/5·25-s + 0.755·28-s + 0.742·29-s + 0.353·32-s − 2.74·34-s + 1.35·35-s + 1/3·36-s + 0.657·37-s − 1.87·41-s − 1.82·43-s − 0.301·44-s + 0.596·45-s − 2/7·49-s − 0.848·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 86397025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86397025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.294613095\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.294613095\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 154 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 166 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−8.072030087044854451712609832327, −7.62926153762886216492375361216, −7.30006999382417069963732032153, −7.17228407899312398049856832382, −6.45363616445473831613587675061, −6.30475046427185450448657535679, −5.92897779151536536843196476874, −5.22545348697061647241077523690, −5.19377388367535862884396994286, −5.01938373227901756208932977384, −4.53401267986844259817562140131, −4.05630227843943496546017351971, −3.45003488722167814650762971944, −3.22219061005423894264054353909, −2.72503746672991041715810033517, −1.96980343936148011121971291438, −1.93781508675787687574555098567, −1.35502173047735544451234590375, −0.952888085064269769201358610919, −0.53629556289382694327575866400,
0.53629556289382694327575866400, 0.952888085064269769201358610919, 1.35502173047735544451234590375, 1.93781508675787687574555098567, 1.96980343936148011121971291438, 2.72503746672991041715810033517, 3.22219061005423894264054353909, 3.45003488722167814650762971944, 4.05630227843943496546017351971, 4.53401267986844259817562140131, 5.01938373227901756208932977384, 5.19377388367535862884396994286, 5.22545348697061647241077523690, 5.92897779151536536843196476874, 6.30475046427185450448657535679, 6.45363616445473831613587675061, 7.17228407899312398049856832382, 7.30006999382417069963732032153, 7.62926153762886216492375361216, 8.072030087044854451712609832327
