| L(s) = 1 | + 2·3-s − 7-s + 3·9-s − 8·19-s − 2·21-s − 6·25-s + 4·27-s + 12·29-s − 16·31-s − 20·37-s − 16·47-s + 49-s + 12·53-s − 16·57-s + 8·59-s − 3·63-s − 12·75-s + 5·81-s + 8·83-s + 24·87-s − 32·93-s + 28·109-s − 40·111-s − 28·113-s − 22·121-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.377·7-s + 9-s − 1.83·19-s − 0.436·21-s − 6/5·25-s + 0.769·27-s + 2.22·29-s − 2.87·31-s − 3.28·37-s − 2.33·47-s + 1/7·49-s + 1.64·53-s − 2.11·57-s + 1.04·59-s − 0.377·63-s − 1.38·75-s + 5/9·81-s + 0.878·83-s + 2.57·87-s − 3.31·93-s + 2.68·109-s − 3.79·111-s − 2.63·113-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 395136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 395136 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( 1 + T \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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
−8.653085291896665921137704810205, −7.984082677547606361405691055111, −7.65994621019403063529610936786, −7.01766097550054006668180221905, −6.57007226264860504077210553618, −6.42136658293515809263418102895, −5.31812839176075303563910872979, −5.25019213638990204722565722077, −4.34225477592797599149198289952, −3.76939099555358992903176811916, −3.56921408081238054859871707771, −2.77867908850904344408617794435, −2.06515692945354055956624655608, −1.66936553538445934358646144492, 0,
1.66936553538445934358646144492, 2.06515692945354055956624655608, 2.77867908850904344408617794435, 3.56921408081238054859871707771, 3.76939099555358992903176811916, 4.34225477592797599149198289952, 5.25019213638990204722565722077, 5.31812839176075303563910872979, 6.42136658293515809263418102895, 6.57007226264860504077210553618, 7.01766097550054006668180221905, 7.65994621019403063529610936786, 7.984082677547606361405691055111, 8.653085291896665921137704810205
