| L(s) = 1 | + 2-s − 3-s + 4-s + 2·5-s − 6-s + 2·7-s + 3·8-s − 9-s + 2·10-s − 12-s − 5·13-s + 2·14-s − 2·15-s + 16-s + 5·17-s − 18-s + 6·19-s + 2·20-s − 2·21-s − 2·23-s − 3·24-s + 3·25-s − 5·26-s + 2·28-s − 29-s − 2·30-s − 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s + 0.755·7-s + 1.06·8-s − 1/3·9-s + 0.632·10-s − 0.288·12-s − 1.38·13-s + 0.534·14-s − 0.516·15-s + 1/4·16-s + 1.21·17-s − 0.235·18-s + 1.37·19-s + 0.447·20-s − 0.436·21-s − 0.417·23-s − 0.612·24-s + 3/5·25-s − 0.980·26-s + 0.377·28-s − 0.185·29-s − 0.365·30-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17935225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17935225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.484114303\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.484114303\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 - T - p T^{3} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 5 T + 28 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 5 T + 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + T + 20 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 2 T + 66 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 10 T + 94 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 5 T + 62 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2 T + 90 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 6 T - 22 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 9 T + 174 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 6 T + 170 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 9 T + 108 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
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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
−8.307434696980645290466484928672, −8.076474363696145274866521427410, −7.75146117261382608314551266218, −7.57875859219602155072550103766, −6.98777578854655369240446679217, −6.76653129177464418176801902683, −6.26361573191413970751500468934, −5.93863727584563029726990417403, −5.42162799028890236171573605504, −5.27046477838754737738577509813, −4.89911789550624531718358182839, −4.74543113158459742090226715029, −4.13052708167918412426827697502, −3.61443848098398586797730894220, −3.18572319067969455754605102762, −2.53933847439655050534891359737, −2.38404939436388091178322230455, −1.52205660299310197849812197677, −1.46367428224501044470697341629, −0.53890190040846363891927657333,
0.53890190040846363891927657333, 1.46367428224501044470697341629, 1.52205660299310197849812197677, 2.38404939436388091178322230455, 2.53933847439655050534891359737, 3.18572319067969455754605102762, 3.61443848098398586797730894220, 4.13052708167918412426827697502, 4.74543113158459742090226715029, 4.89911789550624531718358182839, 5.27046477838754737738577509813, 5.42162799028890236171573605504, 5.93863727584563029726990417403, 6.26361573191413970751500468934, 6.76653129177464418176801902683, 6.98777578854655369240446679217, 7.57875859219602155072550103766, 7.75146117261382608314551266218, 8.076474363696145274866521427410, 8.307434696980645290466484928672
