| L(s) = 1 | + 2·2-s − 3-s + 2·4-s − 5-s − 2·6-s − 2·9-s − 2·10-s − 2·12-s + 3·13-s + 15-s − 4·16-s − 4·18-s − 2·20-s + 25-s + 6·26-s + 5·27-s + 2·30-s − 2·31-s − 8·32-s − 4·36-s − 12·37-s − 3·39-s − 3·41-s + 3·43-s + 2·45-s + 4·48-s − 10·49-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s − 0.447·5-s − 0.816·6-s − 2/3·9-s − 0.632·10-s − 0.577·12-s + 0.832·13-s + 0.258·15-s − 16-s − 0.942·18-s − 0.447·20-s + 1/5·25-s + 1.17·26-s + 0.962·27-s + 0.365·30-s − 0.359·31-s − 1.41·32-s − 2/3·36-s − 1.97·37-s − 0.480·39-s − 0.468·41-s + 0.457·43-s + 0.298·45-s + 0.577·48-s − 1.42·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 3 | $C_2$ | \( 1 + T + p T^{2} \) |
| 5 | $C_1$ | \( 1 + T \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 21 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 15 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 95 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 100 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 176 T^{2} + p^{2} T^{4} \) |
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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
−8.722339061279030502026217557080, −8.326625773390273821916688279470, −7.86104944075813803200111539928, −6.98306956144003059227349427373, −6.80029561603651451983469425798, −6.20925779024294472317126229619, −5.74294810853403922092955208519, −5.32290141168043296240714248598, −4.84739726754712687571159778127, −4.31296215270085071722819888265, −3.57959421488730987783136762088, −3.31251437718625774347934014906, −2.57128120301065922656290658012, −1.55179478107478001093121490033, 0,
1.55179478107478001093121490033, 2.57128120301065922656290658012, 3.31251437718625774347934014906, 3.57959421488730987783136762088, 4.31296215270085071722819888265, 4.84739726754712687571159778127, 5.32290141168043296240714248598, 5.74294810853403922092955208519, 6.20925779024294472317126229619, 6.80029561603651451983469425798, 6.98306956144003059227349427373, 7.86104944075813803200111539928, 8.326625773390273821916688279470, 8.722339061279030502026217557080
