L(s) = 1 | + 2·2-s + 3·4-s + 6·7-s + 4·8-s − 9-s + 4·13-s + 12·14-s + 5·16-s − 2·18-s + 8·26-s + 18·28-s + 10·29-s + 6·32-s − 3·36-s − 4·37-s + 6·47-s + 13·49-s + 12·52-s + 24·56-s + 20·58-s − 26·61-s − 6·63-s + 7·64-s + 26·67-s − 4·72-s − 12·73-s − 8·74-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 3/2·4-s + 2.26·7-s + 1.41·8-s − 1/3·9-s + 1.10·13-s + 3.20·14-s + 5/4·16-s − 0.471·18-s + 1.56·26-s + 3.40·28-s + 1.85·29-s + 1.06·32-s − 1/2·36-s − 0.657·37-s + 0.875·47-s + 13/7·49-s + 1.66·52-s + 3.20·56-s + 2.62·58-s − 3.32·61-s − 0.755·63-s + 7/8·64-s + 3.17·67-s − 0.471·72-s − 1.40·73-s − 0.929·74-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(10.26715829\) |
\(L(\frac12)\) |
\(\approx\) |
\(10.26715829\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
good | 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 105 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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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
−9.174083056813021041848900294074, −8.900708294718218593908172421737, −8.362753038381910379653896399934, −8.221386122725656731195566504438, −7.84962674250234991788385617270, −7.33189067866850681642682591467, −7.05557250673826687196833987821, −6.41317365972876743535271740190, −5.95174744379117662786101475525, −5.91875119374335620317214570393, −5.05038657882760264012725781375, −4.98319389822163979942693905356, −4.53750356151393784038598312343, −4.27924155719094403777257246368, −3.44662618665729491112720039250, −3.35178702918451943111584424923, −2.49153215459591138965778677445, −2.09336847200769477098002337447, −1.45491289552526550486262302402, −1.02595622375540356416283476511,
1.02595622375540356416283476511, 1.45491289552526550486262302402, 2.09336847200769477098002337447, 2.49153215459591138965778677445, 3.35178702918451943111584424923, 3.44662618665729491112720039250, 4.27924155719094403777257246368, 4.53750356151393784038598312343, 4.98319389822163979942693905356, 5.05038657882760264012725781375, 5.91875119374335620317214570393, 5.95174744379117662786101475525, 6.41317365972876743535271740190, 7.05557250673826687196833987821, 7.33189067866850681642682591467, 7.84962674250234991788385617270, 8.221386122725656731195566504438, 8.362753038381910379653896399934, 8.900708294718218593908172421737, 9.174083056813021041848900294074