| L(s) = 1 | + 2-s + 4-s − 6·7-s + 8-s + 9-s − 6·14-s + 16-s + 3·17-s + 18-s + 2·23-s + 2·25-s − 6·28-s + 5·31-s + 32-s + 3·34-s + 36-s + 2·41-s + 2·46-s + 2·47-s + 24·49-s + 2·50-s − 6·56-s + 5·62-s − 6·63-s + 64-s + 3·68-s + 14·71-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 2.26·7-s + 0.353·8-s + 1/3·9-s − 1.60·14-s + 1/4·16-s + 0.727·17-s + 0.235·18-s + 0.417·23-s + 2/5·25-s − 1.13·28-s + 0.898·31-s + 0.176·32-s + 0.514·34-s + 1/6·36-s + 0.312·41-s + 0.294·46-s + 0.291·47-s + 24/7·49-s + 0.282·50-s − 0.801·56-s + 0.635·62-s − 0.755·63-s + 1/8·64-s + 0.363·68-s + 1.66·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 137088 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137088 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.954600206\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.954600206\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 4 T + p T^{2} ) \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 28 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 55 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−9.376675349388470905263396720582, −9.125912359700064549700904211254, −8.261252984654669490871630907111, −7.87281028269522815042887086811, −7.06094840442657991943731948204, −6.80684856559567173711128671950, −6.43787958244406512004604789637, −5.82826125714962227450636348039, −5.39962439255687162302918195100, −4.66996886166900370420282081890, −3.94095954000428593390155572600, −3.47287570484200550918081604769, −2.95870754230524704356650848390, −2.31765577141856902353579660321, −0.871192218148406437780779340427,
0.871192218148406437780779340427, 2.31765577141856902353579660321, 2.95870754230524704356650848390, 3.47287570484200550918081604769, 3.94095954000428593390155572600, 4.66996886166900370420282081890, 5.39962439255687162302918195100, 5.82826125714962227450636348039, 6.43787958244406512004604789637, 6.80684856559567173711128671950, 7.06094840442657991943731948204, 7.87281028269522815042887086811, 8.261252984654669490871630907111, 9.125912359700064549700904211254, 9.376675349388470905263396720582
