| L(s) = 1 | − 2·2-s + 2·4-s + 2·13-s − 4·16-s + 6·17-s − 4·26-s + 8·32-s − 12·34-s + 14·37-s + 16·41-s + 4·52-s + 18·53-s + 24·61-s − 8·64-s + 12·68-s + 22·73-s − 28·74-s − 32·82-s − 26·97-s − 4·101-s − 36·106-s − 2·113-s + 22·121-s − 48·122-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 0.554·13-s − 16-s + 1.45·17-s − 0.784·26-s + 1.41·32-s − 2.05·34-s + 2.30·37-s + 2.49·41-s + 0.554·52-s + 2.47·53-s + 3.07·61-s − 64-s + 1.45·68-s + 2.57·73-s − 3.25·74-s − 3.53·82-s − 2.63·97-s − 0.398·101-s − 3.49·106-s − 0.188·113-s + 2·121-s − 4.34·122-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.226057029\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.226057029\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−9.996890277123390682354009991054, −9.958326190281891474273146774728, −9.393580817275264422290883811800, −9.284883462273930079427006954417, −8.518783887235664801314418914145, −8.308463138288762191516742471610, −7.87422519838173383075382223527, −7.59712923891881673015669410983, −6.95771576889232062784850323731, −6.77522288810702706279349393391, −5.92918359433554534305807609298, −5.74112646552689243256206578988, −5.18908606733813630322571220897, −4.39892952677900730560774447126, −4.01029074418267914517628519757, −3.46835915504507015116038036725, −2.44730792546244114735843260550, −2.35899830498500443573934550327, −0.992525463155269707789615012909, −0.945154214135935140126786809495,
0.945154214135935140126786809495, 0.992525463155269707789615012909, 2.35899830498500443573934550327, 2.44730792546244114735843260550, 3.46835915504507015116038036725, 4.01029074418267914517628519757, 4.39892952677900730560774447126, 5.18908606733813630322571220897, 5.74112646552689243256206578988, 5.92918359433554534305807609298, 6.77522288810702706279349393391, 6.95771576889232062784850323731, 7.59712923891881673015669410983, 7.87422519838173383075382223527, 8.308463138288762191516742471610, 8.518783887235664801314418914145, 9.284883462273930079427006954417, 9.393580817275264422290883811800, 9.958326190281891474273146774728, 9.996890277123390682354009991054
