L(s) = 1 | − 3·4-s − 6·9-s − 11-s + 4·13-s + 5·16-s + 12·17-s − 8·19-s + 8·23-s + 25-s + 18·36-s − 4·37-s + 4·41-s + 3·44-s − 7·49-s − 12·52-s − 4·53-s − 20·61-s − 3·64-s − 32·67-s − 36·68-s + 16·71-s + 28·73-s + 24·76-s + 27·81-s − 8·83-s − 24·92-s + 6·99-s + ⋯ |
L(s) = 1 | − 3/2·4-s − 2·9-s − 0.301·11-s + 1.10·13-s + 5/4·16-s + 2.91·17-s − 1.83·19-s + 1.66·23-s + 1/5·25-s + 3·36-s − 0.657·37-s + 0.624·41-s + 0.452·44-s − 49-s − 1.66·52-s − 0.549·53-s − 2.56·61-s − 3/8·64-s − 3.90·67-s − 4.36·68-s + 1.89·71-s + 3.27·73-s + 2.75·76-s + 3·81-s − 0.878·83-s − 2.50·92-s + 0.603·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1630475 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1630475 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 + p T^{2} \) |
| 11 | $C_1$ | \( 1 + T \) |
good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 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
−7.976948089097668743343323414075, −7.38697174398530240686785439095, −6.64941346027716297585675692254, −6.16821463973303244938343944630, −5.84607711344828752173687730124, −5.41657340769509440101853785094, −5.09766059609940450216711100862, −4.63496212463286937722213894719, −4.02862499293601628744198277130, −3.32813010597177789495028431337, −3.25423179226253980082861279896, −2.70079657296499886704042361091, −1.59173582864331280223475238377, −0.859950872240661984946655527286, 0,
0.859950872240661984946655527286, 1.59173582864331280223475238377, 2.70079657296499886704042361091, 3.25423179226253980082861279896, 3.32813010597177789495028431337, 4.02862499293601628744198277130, 4.63496212463286937722213894719, 5.09766059609940450216711100862, 5.41657340769509440101853785094, 5.84607711344828752173687730124, 6.16821463973303244938343944630, 6.64941346027716297585675692254, 7.38697174398530240686785439095, 7.976948089097668743343323414075