| L(s) = 1 | + 2·2-s + 2·3-s + 2·4-s + 2·5-s + 4·6-s + 4·8-s + 2·9-s + 4·10-s − 2·11-s + 4·12-s − 4·13-s + 4·15-s + 8·16-s + 4·17-s + 4·18-s − 6·19-s + 4·20-s − 4·22-s + 8·24-s + 2·25-s − 8·26-s − 8·27-s + 12·29-s + 8·30-s + 16·31-s + 8·32-s − 4·33-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 4-s + 0.894·5-s + 1.63·6-s + 1.41·8-s + 2/3·9-s + 1.26·10-s − 0.603·11-s + 1.15·12-s − 1.10·13-s + 1.03·15-s + 2·16-s + 0.970·17-s + 0.942·18-s − 1.37·19-s + 0.894·20-s − 0.852·22-s + 1.63·24-s + 2/5·25-s − 1.56·26-s − 1.53·27-s + 2.22·29-s + 1.46·30-s + 2.87·31-s + 1.41·32-s − 0.696·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(14.70375117\) |
| \(L(\frac12)\) |
\(\approx\) |
\(14.70375117\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^3$ | \( 1 - 2 T + 2 T^{2} + 8 T^{3} - 17 T^{4} + 8 p T^{5} + 2 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} ) \) |
| 11 | $C_2^3$ | \( 1 + 2 T + 2 T^{2} - 40 T^{3} - 161 T^{4} - 40 p T^{5} + 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^3$ | \( 1 + 6 T + 18 T^{2} - 120 T^{3} - 721 T^{4} - 120 p T^{5} + 18 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^3$ | \( 1 + 10 T^{2} - 429 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^3$ | \( 1 + 6 T + 18 T^{2} - 336 T^{3} - 2377 T^{4} - 336 p T^{5} + 18 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 14 T + 143 T^{2} - 14 p T^{3} + p^{2} T^{4} )( 1 + 4 T - 37 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \) |
| 59 | $C_2^3$ | \( 1 - 6 T + 18 T^{2} + 600 T^{3} - 5281 T^{4} + 600 p T^{5} + 18 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^3$ | \( 1 - 18 T + 162 T^{2} - 720 T^{3} + 2759 T^{4} - 720 p T^{5} + 162 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2^3$ | \( 1 - 10 T + 50 T^{2} + 840 T^{3} - 8689 T^{4} + 840 p T^{5} + 50 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 42 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^3$ | \( 1 + 130 T^{2} + 11571 T^{4} + 130 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 + 162 T^{2} + 18323 T^{4} + 162 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.47302820336999712822516299098, −7.02353804777793697501279000603, −6.89111219209469923537613511063, −6.82227595370972107388549556394, −6.32087354075610996175627755323, −6.26294337375910027767456182882, −5.86351615621741669134261876481, −5.84135120913581840798836824872, −5.38401274092848588805536387552, −5.07311198676537972605702712708, −5.07158173708198933825912680102, −4.76521228865707865587748470639, −4.65042219647258950486118344057, −4.06390301658789693997804727006, −3.91567929994605170749863323552, −3.86174722923435438824818566210, −3.65876892468198362399949278965, −2.77174750763484560347845687918, −2.76588440608937388935055441829, −2.75791260802413482393592779634, −2.42631594368427233076528805643, −2.03398092580905579983917475861, −1.63568432097551402479036568706, −1.20948754995675415169234022996, −0.68936197981041677942092310878,
0.68936197981041677942092310878, 1.20948754995675415169234022996, 1.63568432097551402479036568706, 2.03398092580905579983917475861, 2.42631594368427233076528805643, 2.75791260802413482393592779634, 2.76588440608937388935055441829, 2.77174750763484560347845687918, 3.65876892468198362399949278965, 3.86174722923435438824818566210, 3.91567929994605170749863323552, 4.06390301658789693997804727006, 4.65042219647258950486118344057, 4.76521228865707865587748470639, 5.07158173708198933825912680102, 5.07311198676537972605702712708, 5.38401274092848588805536387552, 5.84135120913581840798836824872, 5.86351615621741669134261876481, 6.26294337375910027767456182882, 6.32087354075610996175627755323, 6.82227595370972107388549556394, 6.89111219209469923537613511063, 7.02353804777793697501279000603, 7.47302820336999712822516299098
