| L(s) = 1 | − 3-s + 4-s − 4·7-s + 3·9-s − 3·11-s − 12-s + 5·13-s − 3·16-s + 3·17-s + 8·19-s + 4·21-s + 9·23-s + 10·25-s − 8·27-s − 4·28-s + 3·29-s − 7·31-s + 3·33-s + 3·36-s + 3·37-s − 5·39-s − 3·41-s − 7·43-s − 3·44-s + 3·47-s + 3·48-s + 9·49-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/2·4-s − 1.51·7-s + 9-s − 0.904·11-s − 0.288·12-s + 1.38·13-s − 3/4·16-s + 0.727·17-s + 1.83·19-s + 0.872·21-s + 1.87·23-s + 2·25-s − 1.53·27-s − 0.755·28-s + 0.557·29-s − 1.25·31-s + 0.522·33-s + 1/2·36-s + 0.493·37-s − 0.800·39-s − 0.468·41-s − 1.06·43-s − 0.452·44-s + 0.437·47-s + 0.433·48-s + 9/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17689 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17689 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9895883689\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9895883689\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 19 | $C_2$ | \( 1 - 8 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 3 T + 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 3 T + 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 3 T + 40 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 7 T + 6 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 3 T + 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 9 T + 88 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 21 T + 218 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 146 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 134 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 15 T + 136 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - T - 96 T^{2} - p T^{3} + p^{2} T^{4} \) |
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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
−13.68470261771557086333796021964, −13.01846002779985000642267892509, −12.64534044218936075556824669031, −11.97958649163151395994016472701, −11.46107558547595672236187016168, −10.87730694452263436171332654717, −10.57333679883320162077858131777, −10.00547133047382846129366205943, −9.292859311003101820292074172861, −9.101287095415886157491507741689, −8.187147792334698397167219724853, −7.17622526206904361731278235053, −7.14925514281958595949169461262, −6.54144550217376443552867707505, −5.71899565014024450614597841021, −5.31440760399197553967929304769, −4.41864517465389456646413945425, −3.17504788315366696936069515708, −3.10285126953531171079661674310, −1.24442070893612661661689321331,
1.24442070893612661661689321331, 3.10285126953531171079661674310, 3.17504788315366696936069515708, 4.41864517465389456646413945425, 5.31440760399197553967929304769, 5.71899565014024450614597841021, 6.54144550217376443552867707505, 7.14925514281958595949169461262, 7.17622526206904361731278235053, 8.187147792334698397167219724853, 9.101287095415886157491507741689, 9.292859311003101820292074172861, 10.00547133047382846129366205943, 10.57333679883320162077858131777, 10.87730694452263436171332654717, 11.46107558547595672236187016168, 11.97958649163151395994016472701, 12.64534044218936075556824669031, 13.01846002779985000642267892509, 13.68470261771557086333796021964
