| L(s) = 1 | − 2·3-s + 5-s − 2·7-s + 3·9-s − 4·11-s − 2·13-s − 2·15-s − 7·19-s + 4·21-s − 3·23-s − 25-s − 4·27-s − 3·29-s + 31-s + 8·33-s − 2·35-s + 2·37-s + 4·39-s + 6·41-s − 5·43-s + 3·45-s + 7·47-s + 3·49-s + 5·53-s − 4·55-s + 14·57-s + 16·59-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.447·5-s − 0.755·7-s + 9-s − 1.20·11-s − 0.554·13-s − 0.516·15-s − 1.60·19-s + 0.872·21-s − 0.625·23-s − 1/5·25-s − 0.769·27-s − 0.557·29-s + 0.179·31-s + 1.39·33-s − 0.338·35-s + 0.328·37-s + 0.640·39-s + 0.937·41-s − 0.762·43-s + 0.447·45-s + 1.02·47-s + 3/7·49-s + 0.686·53-s − 0.539·55-s + 1.85·57-s + 2.08·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19079424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19079424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8288082233\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8288082233\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 + 7 T + 42 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 3 T + 40 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 3 T + 52 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - T + 54 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 6 T + 58 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 5 T + 18 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 7 T + 98 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 5 T + 104 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 + 14 T + 150 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 6 T + 118 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - T - 60 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 5 T + 90 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 5 T + 98 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 9 T + 190 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 29 T + 396 T^{2} - 29 p T^{3} + p^{2} T^{4} \) |
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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
−8.480223720048067934265182936939, −8.249473605685829877364603480190, −7.59357320273006210296343134822, −7.49601903434926689215679678568, −7.01811843996224926472045534946, −6.67072099661234516399236519480, −6.20773547269953331872120937415, −6.09190148526156743915440064587, −5.54123113914894278657470887588, −5.35114506084774601722254793464, −5.01866825245268575779711208206, −4.36339911723814671209082687899, −4.09777281569580816708423624964, −3.82459633835195297977007273620, −2.97842774283939190974358528708, −2.67937996143861248109210779631, −2.01242170353375668379107677079, −1.94224996286917475886429854434, −0.796109104321511027293201729806, −0.36409580788311828761240336846,
0.36409580788311828761240336846, 0.796109104321511027293201729806, 1.94224996286917475886429854434, 2.01242170353375668379107677079, 2.67937996143861248109210779631, 2.97842774283939190974358528708, 3.82459633835195297977007273620, 4.09777281569580816708423624964, 4.36339911723814671209082687899, 5.01866825245268575779711208206, 5.35114506084774601722254793464, 5.54123113914894278657470887588, 6.09190148526156743915440064587, 6.20773547269953331872120937415, 6.67072099661234516399236519480, 7.01811843996224926472045534946, 7.49601903434926689215679678568, 7.59357320273006210296343134822, 8.249473605685829877364603480190, 8.480223720048067934265182936939
