| L(s) = 1 | + 2·3-s − 4·4-s − 3·9-s − 8·12-s − 4·13-s + 12·16-s + 12·17-s + 12·23-s − 10·25-s − 14·27-s − 12·29-s + 12·36-s − 8·39-s + 16·43-s + 24·48-s − 13·49-s + 24·51-s + 16·52-s − 6·53-s + 16·61-s − 32·64-s − 48·68-s + 24·69-s − 20·75-s − 20·79-s − 4·81-s − 24·87-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 2·4-s − 9-s − 2.30·12-s − 1.10·13-s + 3·16-s + 2.91·17-s + 2.50·23-s − 2·25-s − 2.69·27-s − 2.22·29-s + 2·36-s − 1.28·39-s + 2.43·43-s + 3.46·48-s − 1.85·49-s + 3.36·51-s + 2.21·52-s − 0.824·53-s + 2.04·61-s − 4·64-s − 5.82·68-s + 2.88·69-s − 2.30·75-s − 2.25·79-s − 4/9·81-s − 2.57·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 231361 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231361 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−9.032345851694468357832029856913, −8.241330915774091826817607809326, −7.932252712474312694875118416259, −7.59911177067371416247669368567, −7.33591996649962500060513363229, −5.97965532485801959040073464192, −5.52163728124810212778032448117, −5.44973416215471530225317007593, −4.84323437999328576885334632783, −3.96753607282613353734743159883, −3.50910294340479626549928321873, −3.22077100740487422350904766970, −2.49170215186160867754656573885, −1.24474461137231690537182545671, 0,
1.24474461137231690537182545671, 2.49170215186160867754656573885, 3.22077100740487422350904766970, 3.50910294340479626549928321873, 3.96753607282613353734743159883, 4.84323437999328576885334632783, 5.44973416215471530225317007593, 5.52163728124810212778032448117, 5.97965532485801959040073464192, 7.33591996649962500060513363229, 7.59911177067371416247669368567, 7.932252712474312694875118416259, 8.241330915774091826817607809326, 9.032345851694468357832029856913
