| L(s) = 1 | − 2·3-s + 4-s − 2·7-s + 9-s − 2·12-s − 8·13-s + 16-s + 4·19-s + 4·21-s + 25-s + 4·27-s − 2·28-s + 10·31-s + 36-s + 2·37-s + 16·39-s − 2·43-s − 2·48-s − 11·49-s − 8·52-s − 8·57-s − 2·61-s − 2·63-s + 64-s − 8·67-s − 32·73-s − 2·75-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/2·4-s − 0.755·7-s + 1/3·9-s − 0.577·12-s − 2.21·13-s + 1/4·16-s + 0.917·19-s + 0.872·21-s + 1/5·25-s + 0.769·27-s − 0.377·28-s + 1.79·31-s + 1/6·36-s + 0.328·37-s + 2.56·39-s − 0.304·43-s − 0.288·48-s − 1.57·49-s − 1.10·52-s − 1.05·57-s − 0.256·61-s − 0.251·63-s + 1/8·64-s − 0.977·67-s − 3.74·73-s − 0.230·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 37 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2} \) |
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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
−7.48472813821623192367156751152, −7.47402466508823037518119617944, −6.73483784989315938735561487128, −6.56330850429489211825831149583, −5.99940287722666102586658741510, −5.73299805716885755231121374807, −5.06620453918957560960032027985, −4.60963444019434721197685153570, −4.59767259681282669603710565857, −3.42767809661089504801834268834, −2.99576716331676506161499712038, −2.62659393830675313979656999849, −1.82989205635176673644713429623, −0.856493313433969578668782918754, 0,
0.856493313433969578668782918754, 1.82989205635176673644713429623, 2.62659393830675313979656999849, 2.99576716331676506161499712038, 3.42767809661089504801834268834, 4.59767259681282669603710565857, 4.60963444019434721197685153570, 5.06620453918957560960032027985, 5.73299805716885755231121374807, 5.99940287722666102586658741510, 6.56330850429489211825831149583, 6.73483784989315938735561487128, 7.47402466508823037518119617944, 7.48472813821623192367156751152
