L(s) = 1 | − 4·3-s + 2·7-s + 6·9-s − 8·19-s − 8·21-s + 25-s + 4·27-s + 12·29-s − 8·31-s + 4·37-s − 12·47-s − 3·49-s − 12·53-s + 32·57-s + 24·59-s + 12·63-s − 4·75-s − 37·81-s + 12·83-s − 48·87-s + 32·93-s + 28·103-s + 4·109-s − 16·111-s − 12·113-s − 22·121-s + 127-s + ⋯ |
L(s) = 1 | − 2.30·3-s + 0.755·7-s + 2·9-s − 1.83·19-s − 1.74·21-s + 1/5·25-s + 0.769·27-s + 2.22·29-s − 1.43·31-s + 0.657·37-s − 1.75·47-s − 3/7·49-s − 1.64·53-s + 4.23·57-s + 3.12·59-s + 1.51·63-s − 0.461·75-s − 4.11·81-s + 1.31·83-s − 5.14·87-s + 3.31·93-s + 2.75·103-s + 0.383·109-s − 1.51·111-s − 1.12·113-s − 2·121-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5025097476\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5025097476\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
good | 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−10.14952772720870679951662239777, −9.324513479586134673187748298856, −8.552217550781204179646223792184, −8.371569178851438535097195720359, −7.72417967231205206440371091917, −6.75938606427748610323373549762, −6.57891116465648258947670054106, −6.16943207782230729958538690385, −5.54097686482462236146375699951, −4.92936609872116028004359556395, −4.78130792717525308450176413839, −4.02348966113043943975885714437, −2.91432418996240576588953142404, −1.84751760735400887993660553283, −0.62674328168244276773408798542,
0.62674328168244276773408798542, 1.84751760735400887993660553283, 2.91432418996240576588953142404, 4.02348966113043943975885714437, 4.78130792717525308450176413839, 4.92936609872116028004359556395, 5.54097686482462236146375699951, 6.16943207782230729958538690385, 6.57891116465648258947670054106, 6.75938606427748610323373549762, 7.72417967231205206440371091917, 8.371569178851438535097195720359, 8.552217550781204179646223792184, 9.324513479586134673187748298856, 10.14952772720870679951662239777