L(s) = 1 | + 2·2-s + 2·4-s − 4·5-s − 8·10-s + 4·13-s − 4·16-s + 10·17-s − 8·20-s + 11·25-s + 8·26-s − 8·32-s + 20·34-s + 14·37-s + 22·50-s + 8·52-s + 10·53-s − 24·61-s − 8·64-s − 16·65-s + 20·68-s + 22·73-s + 28·74-s + 16·80-s − 9·81-s − 40·85-s + 20·89-s + 26·97-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 4-s − 1.78·5-s − 2.52·10-s + 1.10·13-s − 16-s + 2.42·17-s − 1.78·20-s + 11/5·25-s + 1.56·26-s − 1.41·32-s + 3.42·34-s + 2.30·37-s + 3.11·50-s + 1.10·52-s + 1.37·53-s − 3.07·61-s − 64-s − 1.98·65-s + 2.42·68-s + 2.57·73-s + 3.25·74-s + 1.78·80-s − 81-s − 4.33·85-s + 2.11·89-s + 2.63·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.281104873\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.281104873\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 5 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
good | 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
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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
−12.19808920026082555490503275323, −11.83479989647058711874851966046, −11.62520358704486164764320485690, −10.94764304595673512092141508531, −10.67440659049052817001036729708, −9.985301004627063300813341812259, −9.197057706338527608000351424781, −8.893088393591458338606294055737, −7.956562089194112718817643091403, −7.83112531043615601525828886727, −7.46251951013669094485232831439, −6.37087300067550339343483220817, −6.33534823329739839119619728991, −5.37054128530512854989016519804, −5.06211686255983221878751417400, −4.04717583880578270689560921770, −3.99713765636240581374531385895, −3.27607928396725695438596259054, −2.78410593719181404868377758581, −1.05784920675631122175991650457,
1.05784920675631122175991650457, 2.78410593719181404868377758581, 3.27607928396725695438596259054, 3.99713765636240581374531385895, 4.04717583880578270689560921770, 5.06211686255983221878751417400, 5.37054128530512854989016519804, 6.33534823329739839119619728991, 6.37087300067550339343483220817, 7.46251951013669094485232831439, 7.83112531043615601525828886727, 7.956562089194112718817643091403, 8.893088393591458338606294055737, 9.197057706338527608000351424781, 9.985301004627063300813341812259, 10.67440659049052817001036729708, 10.94764304595673512092141508531, 11.62520358704486164764320485690, 11.83479989647058711874851966046, 12.19808920026082555490503275323