L(s) = 1 | − 2·2-s + 4-s + 2·7-s + 4·11-s + 4·13-s − 4·14-s + 16-s − 4·17-s − 8·22-s − 4·23-s − 8·26-s + 2·28-s + 16·29-s + 2·32-s + 8·34-s + 12·37-s − 4·41-s + 8·43-s + 4·44-s + 8·46-s + 8·47-s + 3·49-s + 4·52-s − 16·53-s − 32·58-s − 8·59-s + 12·61-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 1/2·4-s + 0.755·7-s + 1.20·11-s + 1.10·13-s − 1.06·14-s + 1/4·16-s − 0.970·17-s − 1.70·22-s − 0.834·23-s − 1.56·26-s + 0.377·28-s + 2.97·29-s + 0.353·32-s + 1.37·34-s + 1.97·37-s − 0.624·41-s + 1.21·43-s + 0.603·44-s + 1.17·46-s + 1.16·47-s + 3/7·49-s + 0.554·52-s − 2.19·53-s − 4.20·58-s − 1.04·59-s + 1.53·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.350298877\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.350298877\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 2 | $D_{4}$ | \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_4$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 16 T + 162 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 22 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 74 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T - 50 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 126 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 89 | $C_4$ | \( 1 - 12 T + 86 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 12 T + 222 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
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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
−9.285349067717175880142989476131, −9.281241068477553963774117269923, −8.728426403602027105611871205858, −8.683523398064245931358573386170, −8.112904435866549713564142443748, −7.76810097449512895131178544168, −7.62873576403561617131577714023, −6.63093532373398252874187639222, −6.42487303242401377188897156611, −6.31858370100240807587918287960, −5.73240979539694296258335927157, −4.94815588590116147010231314792, −4.51655591641503877926536922441, −4.28872564812333637304099569407, −3.67706109614843316059969298026, −3.06720417886087042793904368961, −2.38002570622359334650757213469, −1.83627792410151497358758155052, −0.891019597558121091142297421618, −0.864988085239673802908259165882,
0.864988085239673802908259165882, 0.891019597558121091142297421618, 1.83627792410151497358758155052, 2.38002570622359334650757213469, 3.06720417886087042793904368961, 3.67706109614843316059969298026, 4.28872564812333637304099569407, 4.51655591641503877926536922441, 4.94815588590116147010231314792, 5.73240979539694296258335927157, 6.31858370100240807587918287960, 6.42487303242401377188897156611, 6.63093532373398252874187639222, 7.62873576403561617131577714023, 7.76810097449512895131178544168, 8.112904435866549713564142443748, 8.683523398064245931358573386170, 8.728426403602027105611871205858, 9.281241068477553963774117269923, 9.285349067717175880142989476131