L(s) = 1 | − 2-s − 4-s + 3·8-s + 2·9-s − 4·11-s − 16-s − 2·18-s + 4·22-s + 4·23-s + 25-s − 16·29-s − 5·32-s − 2·36-s + 4·37-s + 20·43-s + 4·44-s − 4·46-s − 50-s − 16·53-s + 16·58-s + 7·64-s − 4·71-s + 6·72-s − 4·74-s − 4·79-s − 5·81-s − 20·86-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.06·8-s + 2/3·9-s − 1.20·11-s − 1/4·16-s − 0.471·18-s + 0.852·22-s + 0.834·23-s + 1/5·25-s − 2.97·29-s − 0.883·32-s − 1/3·36-s + 0.657·37-s + 3.04·43-s + 0.603·44-s − 0.589·46-s − 0.141·50-s − 2.19·53-s + 2.10·58-s + 7/8·64-s − 0.474·71-s + 0.707·72-s − 0.464·74-s − 0.450·79-s − 5/9·81-s − 2.15·86-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8194902165\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8194902165\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | | \( 1 \) |
good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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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
−8.029653782412982378537803521682, −7.66805685975589411858394461073, −7.40193216570467925869541631815, −7.22887378834846030685258806963, −6.36057421454209703069981314982, −5.87755209309343129382311355099, −5.40559845872245846250404528783, −5.02357509789140542974507494717, −4.40376809050662069066795034403, −4.10264759968764759982618480522, −3.43625960307185717578067514463, −2.78681709343134764202816406902, −2.07166890218215255418826578341, −1.43594864050468805304724050372, −0.49856023005202788766729720945,
0.49856023005202788766729720945, 1.43594864050468805304724050372, 2.07166890218215255418826578341, 2.78681709343134764202816406902, 3.43625960307185717578067514463, 4.10264759968764759982618480522, 4.40376809050662069066795034403, 5.02357509789140542974507494717, 5.40559845872245846250404528783, 5.87755209309343129382311355099, 6.36057421454209703069981314982, 7.22887378834846030685258806963, 7.40193216570467925869541631815, 7.66805685975589411858394461073, 8.029653782412982378537803521682