L(s) = 1 | − 2-s − 2·3-s + 4-s + 2·6-s + 7-s − 8-s + 9-s − 2·12-s − 8·13-s − 14-s + 16-s + 12·17-s − 18-s + 4·19-s − 2·21-s + 2·24-s − 10·25-s + 8·26-s + 4·27-s + 28-s − 12·29-s − 32-s − 12·34-s + 36-s − 4·38-s + 16·39-s + 12·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.577·12-s − 2.21·13-s − 0.267·14-s + 1/4·16-s + 2.91·17-s − 0.235·18-s + 0.917·19-s − 0.436·21-s + 0.408·24-s − 2·25-s + 1.56·26-s + 0.769·27-s + 0.188·28-s − 2.22·29-s − 0.176·32-s − 2.05·34-s + 1/6·36-s − 0.648·38-s + 2.56·39-s + 1.87·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 395136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 395136 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 7 | $C_1$ | \( 1 - T \) |
good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−8.180875231063652135621475257556, −7.80162284551435737726061482141, −7.57571100088867902110310233811, −7.28469638284596991373763433627, −6.62537535772774211952316132348, −5.90261845370297978971697518712, −5.57928681742950427486583645839, −5.25497073120067854106238938582, −4.91908037573729106874161398618, −3.89710194770158862288870018158, −3.45702381146523821097408540658, −2.62516665423511559069595041672, −1.92334915190435393939689868597, −1.03037076519213818080472588034, 0,
1.03037076519213818080472588034, 1.92334915190435393939689868597, 2.62516665423511559069595041672, 3.45702381146523821097408540658, 3.89710194770158862288870018158, 4.91908037573729106874161398618, 5.25497073120067854106238938582, 5.57928681742950427486583645839, 5.90261845370297978971697518712, 6.62537535772774211952316132348, 7.28469638284596991373763433627, 7.57571100088867902110310233811, 7.80162284551435737726061482141, 8.180875231063652135621475257556