| L(s) = 1 | − 3-s − 6·5-s − 5·7-s + 9-s + 6·15-s − 4·16-s − 2·17-s + 5·21-s + 17·25-s − 27-s + 30·35-s − 2·43-s − 6·45-s − 18·47-s + 4·48-s + 18·49-s + 2·51-s − 16·59-s − 5·63-s + 16·67-s − 17·75-s + 32·79-s + 24·80-s + 81-s + 24·83-s + 12·85-s − 12·89-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 2.68·5-s − 1.88·7-s + 1/3·9-s + 1.54·15-s − 16-s − 0.485·17-s + 1.09·21-s + 17/5·25-s − 0.192·27-s + 5.07·35-s − 0.304·43-s − 0.894·45-s − 2.62·47-s + 0.577·48-s + 18/7·49-s + 0.280·51-s − 2.08·59-s − 0.629·63-s + 1.95·67-s − 1.96·75-s + 3.60·79-s + 2.68·80-s + 1/9·81-s + 2.63·83-s + 1.30·85-s − 1.27·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 477603 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 477603 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 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
−8.082649983692116685518449596562, −7.85582869646283906731190811336, −7.47848778675296589128891353641, −6.77124936049535641641720020484, −6.54381789640621822379739731453, −6.41173993136661837405396202601, −5.42089478131328865490166156321, −4.80004704636396746296312994207, −4.43252527426558787350954315583, −3.81591641511052084615077615666, −3.48189187269012783645528672334, −3.11587306879081144724928834922, −2.15535598087150874841625030639, −0.58876266921858160851854225947, 0,
0.58876266921858160851854225947, 2.15535598087150874841625030639, 3.11587306879081144724928834922, 3.48189187269012783645528672334, 3.81591641511052084615077615666, 4.43252527426558787350954315583, 4.80004704636396746296312994207, 5.42089478131328865490166156321, 6.41173993136661837405396202601, 6.54381789640621822379739731453, 6.77124936049535641641720020484, 7.47848778675296589128891353641, 7.85582869646283906731190811336, 8.082649983692116685518449596562
