| L(s) = 1 | + 3·2-s + 3·4-s − 4·7-s − 3·8-s + 2·9-s + 2·11-s − 12·14-s − 13·16-s + 6·18-s + 6·22-s + 2·23-s + 3·25-s − 12·28-s + 3·29-s − 15·32-s + 6·36-s − 6·37-s + 2·43-s + 6·44-s + 6·46-s + 9·49-s + 9·50-s − 3·53-s + 12·56-s + 9·58-s − 8·63-s + 3·64-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 3/2·4-s − 1.51·7-s − 1.06·8-s + 2/3·9-s + 0.603·11-s − 3.20·14-s − 3.25·16-s + 1.41·18-s + 1.27·22-s + 0.417·23-s + 3/5·25-s − 2.26·28-s + 0.557·29-s − 2.65·32-s + 36-s − 0.986·37-s + 0.304·43-s + 0.904·44-s + 0.884·46-s + 9/7·49-s + 1.27·50-s − 0.412·53-s + 1.60·56-s + 1.18·58-s − 1.00·63-s + 3/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1254449 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1254449 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.598630653\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.598630653\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 25601 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 174 T + p T^{2} ) \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 - p T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 24 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 107 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 52 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 176 T^{2} + p^{2} T^{4} \) |
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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
−7.85608478410137696148394872786, −7.13960266120642064419187632962, −6.94782506548202492427014875720, −6.48968999335504325611301690222, −6.10138245761033052348169403147, −5.78701343800836733317418195861, −5.19466984801516220817140717785, −4.82581872098661574494861667594, −4.30854372640905205190606444206, −3.96864991644025912380396976711, −3.51145500070149500323227919849, −3.00044842939720538946939464868, −2.77977321107187284960003855893, −1.74702565684352941571986472969, −0.58958466115059773794067561976,
0.58958466115059773794067561976, 1.74702565684352941571986472969, 2.77977321107187284960003855893, 3.00044842939720538946939464868, 3.51145500070149500323227919849, 3.96864991644025912380396976711, 4.30854372640905205190606444206, 4.82581872098661574494861667594, 5.19466984801516220817140717785, 5.78701343800836733317418195861, 6.10138245761033052348169403147, 6.48968999335504325611301690222, 6.94782506548202492427014875720, 7.13960266120642064419187632962, 7.85608478410137696148394872786
