| L(s) = 1 | + 2·2-s + 3·4-s − 2·7-s + 4·8-s − 5·9-s − 6·11-s − 4·14-s + 5·16-s − 10·18-s − 12·22-s − 12·23-s − 6·28-s + 6·32-s − 15·36-s − 4·37-s + 8·43-s − 18·44-s − 24·46-s − 3·49-s − 12·53-s − 8·56-s + 10·63-s + 7·64-s + 26·67-s + 24·71-s − 20·72-s − 8·74-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 0.755·7-s + 1.41·8-s − 5/3·9-s − 1.80·11-s − 1.06·14-s + 5/4·16-s − 2.35·18-s − 2.55·22-s − 2.50·23-s − 1.13·28-s + 1.06·32-s − 5/2·36-s − 0.657·37-s + 1.21·43-s − 2.71·44-s − 3.53·46-s − 3/7·49-s − 1.64·53-s − 1.06·56-s + 1.25·63-s + 7/8·64-s + 3.17·67-s + 2.84·71-s − 2.35·72-s − 0.929·74-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−9.270518295639118734595954602488, −8.409810000821400938912682522137, −8.071729218562685355881418106440, −7.82315988608083832795544283712, −7.03565158477587046360278529923, −6.35306147256574072432179047412, −6.08705351503109341254138790112, −5.38603857261488641210316347351, −5.34945783064988058695168525193, −4.49973313906049110134055340472, −3.70222391402935277847702964230, −3.29699975126484092788987476482, −2.50196990522414289793307495614, −2.23496341365700158267997210371, 0,
2.23496341365700158267997210371, 2.50196990522414289793307495614, 3.29699975126484092788987476482, 3.70222391402935277847702964230, 4.49973313906049110134055340472, 5.34945783064988058695168525193, 5.38603857261488641210316347351, 6.08705351503109341254138790112, 6.35306147256574072432179047412, 7.03565158477587046360278529923, 7.82315988608083832795544283712, 8.071729218562685355881418106440, 8.409810000821400938912682522137, 9.270518295639118734595954602488
