| L(s) = 1 | − 2·2-s + 2·3-s + 3·4-s − 3·5-s − 4·6-s + 2·7-s − 4·8-s + 3·9-s + 6·10-s − 11-s + 6·12-s − 4·14-s − 6·15-s + 5·16-s − 17-s − 6·18-s − 3·19-s − 9·20-s + 4·21-s + 2·22-s − 7·23-s − 8·24-s + 25-s + 4·27-s + 6·28-s + 29-s + 12·30-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s − 1.34·5-s − 1.63·6-s + 0.755·7-s − 1.41·8-s + 9-s + 1.89·10-s − 0.301·11-s + 1.73·12-s − 1.06·14-s − 1.54·15-s + 5/4·16-s − 0.242·17-s − 1.41·18-s − 0.688·19-s − 2.01·20-s + 0.872·21-s + 0.426·22-s − 1.45·23-s − 1.63·24-s + 1/5·25-s + 0.769·27-s + 1.13·28-s + 0.185·29-s + 2.19·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50381604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50381604 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + T + 18 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 7 T + 54 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - T + 20 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 11 T + 100 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 74 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 17 T + 154 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 12 T + 86 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - T + 84 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 6 T + 126 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 3 T + 144 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 14 T + 190 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 26 T + 318 T^{2} + 26 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 4 T - 74 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−7.892160179088330769831605316233, −7.70948632339476216024453168914, −7.12107149984230423469997921001, −7.07830379124693351335360437209, −6.59166939091016967318563217950, −6.27250817881240225207468998758, −5.60434973517244919085179955025, −5.46791791559510914172237761954, −4.72114635013365308217730092992, −4.34793185877606601802872734577, −4.04726740770731862203467167078, −3.84122513373606083133920354406, −3.10588822765870556371006148251, −2.93744840379310970072257689417, −2.25419486317910645332168112239, −2.15883048094151661725502822760, −1.41426474856606762153725323271, −1.14261155935031524464995188653, 0, 0,
1.14261155935031524464995188653, 1.41426474856606762153725323271, 2.15883048094151661725502822760, 2.25419486317910645332168112239, 2.93744840379310970072257689417, 3.10588822765870556371006148251, 3.84122513373606083133920354406, 4.04726740770731862203467167078, 4.34793185877606601802872734577, 4.72114635013365308217730092992, 5.46791791559510914172237761954, 5.60434973517244919085179955025, 6.27250817881240225207468998758, 6.59166939091016967318563217950, 7.07830379124693351335360437209, 7.12107149984230423469997921001, 7.70948632339476216024453168914, 7.892160179088330769831605316233
