| L(s) = 1 | − 2·3-s − 4·4-s − 2·5-s + 3·9-s − 3·11-s + 8·12-s + 4·15-s + 12·16-s + 8·20-s + 6·23-s − 7·25-s − 4·27-s + 8·31-s + 6·33-s − 12·36-s + 2·37-s + 12·44-s − 6·45-s − 2·47-s − 24·48-s − 5·49-s + 20·53-s + 6·55-s − 20·59-s − 16·60-s − 32·64-s + 8·67-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 2·4-s − 0.894·5-s + 9-s − 0.904·11-s + 2.30·12-s + 1.03·15-s + 3·16-s + 1.78·20-s + 1.25·23-s − 7/5·25-s − 0.769·27-s + 1.43·31-s + 1.04·33-s − 2·36-s + 0.328·37-s + 1.80·44-s − 0.894·45-s − 0.291·47-s − 3.46·48-s − 5/7·49-s + 2.74·53-s + 0.809·55-s − 2.60·59-s − 2.06·60-s − 4·64-s + 0.977·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2405601 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2405601 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| 47 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
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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
−7.40808727808626853293415458610, −7.30736849693508971465380420611, −6.49866775716468763568864632059, −5.84394202900095699724791855542, −5.83486052932247599921983260982, −5.20875822149789983728756765998, −4.75349830419579613526576050031, −4.51796599257777476235432042557, −4.22464308937771834849473683659, −3.43936351267125546314367367616, −3.31847810114563157193588872502, −2.34041331734360102022304929995, −1.26959641434782389797699375762, −0.63762723395673395009540255362, 0,
0.63762723395673395009540255362, 1.26959641434782389797699375762, 2.34041331734360102022304929995, 3.31847810114563157193588872502, 3.43936351267125546314367367616, 4.22464308937771834849473683659, 4.51796599257777476235432042557, 4.75349830419579613526576050031, 5.20875822149789983728756765998, 5.83486052932247599921983260982, 5.84394202900095699724791855542, 6.49866775716468763568864632059, 7.30736849693508971465380420611, 7.40808727808626853293415458610
