| L(s) = 1 | + 2-s − 4-s + 2·5-s + 3·7-s − 3·8-s − 3·9-s + 2·10-s + 11-s + 3·13-s + 3·14-s − 16-s − 3·18-s − 2·20-s + 22-s + 2·25-s + 3·26-s − 3·28-s + 3·31-s + 5·32-s + 6·35-s + 3·36-s − 6·40-s − 12·43-s − 44-s − 6·45-s − 12·47-s + 2·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.894·5-s + 1.13·7-s − 1.06·8-s − 9-s + 0.632·10-s + 0.301·11-s + 0.832·13-s + 0.801·14-s − 1/4·16-s − 0.707·18-s − 0.447·20-s + 0.213·22-s + 2/5·25-s + 0.588·26-s − 0.566·28-s + 0.538·31-s + 0.883·32-s + 1.01·35-s + 1/2·36-s − 0.948·40-s − 1.82·43-s − 0.150·44-s − 0.894·45-s − 1.75·47-s + 2/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.564364528\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.564364528\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( 1 - 3 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 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 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 27 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 27 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 80 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 84 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 68 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 103 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 97 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 166 T^{2} + p^{2} T^{4} \) |
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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
−11.29018342027031247163198659167, −10.59108746600224542497941176388, −10.01990235037557200090829106817, −9.380205880740515291188750413260, −8.866487127923862992769879045338, −8.307653760889788843377949938744, −8.056248543667434837627800020752, −6.88480246084110664183288432293, −6.19136463312070460860119042769, −5.80344882888245973012737017581, −5.05202822379673736476693870517, −4.68177149673329685613825793712, −3.66003826158809657674050062347, −2.89689599451033385628864859966, −1.65288184070394826328171488252,
1.65288184070394826328171488252, 2.89689599451033385628864859966, 3.66003826158809657674050062347, 4.68177149673329685613825793712, 5.05202822379673736476693870517, 5.80344882888245973012737017581, 6.19136463312070460860119042769, 6.88480246084110664183288432293, 8.056248543667434837627800020752, 8.307653760889788843377949938744, 8.866487127923862992769879045338, 9.380205880740515291188750413260, 10.01990235037557200090829106817, 10.59108746600224542497941176388, 11.29018342027031247163198659167
