| L(s) = 1 | − 2·2-s − 4·3-s − 6·5-s + 8·6-s − 2·7-s + 4·8-s + 7·9-s + 12·10-s − 8·11-s − 8·13-s + 4·14-s + 24·15-s − 4·16-s − 14·18-s − 4·19-s + 8·21-s + 16·22-s − 6·23-s − 16·24-s + 18·25-s + 16·26-s − 4·27-s + 7·29-s − 48·30-s − 2·31-s + 32·33-s + 12·35-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 2.30·3-s − 2.68·5-s + 3.26·6-s − 0.755·7-s + 1.41·8-s + 7/3·9-s + 3.79·10-s − 2.41·11-s − 2.21·13-s + 1.06·14-s + 6.19·15-s − 16-s − 3.29·18-s − 0.917·19-s + 1.74·21-s + 3.41·22-s − 1.25·23-s − 3.26·24-s + 18/5·25-s + 3.13·26-s − 0.769·27-s + 1.29·29-s − 8.76·30-s − 0.359·31-s + 5.57·33-s + 2.02·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39701 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39701 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 29 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 6 T + p T^{2} ) \) |
| 37 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 3 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 9 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
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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
−15.9321465971, −15.6038578732, −15.0793819607, −14.6177703295, −13.6592191581, −12.9583864139, −12.7039531267, −12.0840505686, −11.7675102999, −11.7573247228, −10.7751381625, −10.5559389095, −10.0913418946, −9.93309835361, −8.88353970489, −8.13243112487, −8.01433080787, −7.66019523297, −6.87039121695, −6.55092616937, −5.32164607910, −5.00317001401, −4.67964820972, −3.84581812695, −2.76581983431, 0, 0, 0,
2.76581983431, 3.84581812695, 4.67964820972, 5.00317001401, 5.32164607910, 6.55092616937, 6.87039121695, 7.66019523297, 8.01433080787, 8.13243112487, 8.88353970489, 9.93309835361, 10.0913418946, 10.5559389095, 10.7751381625, 11.7573247228, 11.7675102999, 12.0840505686, 12.7039531267, 12.9583864139, 13.6592191581, 14.6177703295, 15.0793819607, 15.6038578732, 15.9321465971
