| L(s) = 1 | + 2·3-s − 2·5-s + 3·9-s − 4·13-s − 4·15-s + 4·17-s − 4·23-s + 3·25-s + 4·27-s − 4·29-s − 2·31-s − 12·37-s − 8·39-s − 4·41-s + 16·43-s − 6·45-s − 8·47-s − 8·49-s + 8·51-s − 12·53-s − 4·61-s + 8·65-s + 8·67-s − 8·69-s − 8·71-s − 20·73-s + 6·75-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s + 9-s − 1.10·13-s − 1.03·15-s + 0.970·17-s − 0.834·23-s + 3/5·25-s + 0.769·27-s − 0.742·29-s − 0.359·31-s − 1.97·37-s − 1.28·39-s − 0.624·41-s + 2.43·43-s − 0.894·45-s − 1.16·47-s − 8/7·49-s + 1.12·51-s − 1.64·53-s − 0.512·61-s + 0.992·65-s + 0.977·67-s − 0.963·69-s − 0.949·71-s − 2.34·73-s + 0.692·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55353600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55353600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 31 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 24 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 26 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 56 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 12 T + 104 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T - 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 16 T + 126 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 8 T + 86 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 64 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T + 102 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 144 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 152 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 20 T + 240 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 4 T + 138 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 12 T + 178 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 176 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 20 T + 270 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
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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
−7.70630676768948473668775565710, −7.50462914175399962161902402838, −7.06468057705941292429778841259, −7.04168207828962926948327479357, −6.27388854555689088525312454210, −6.16126241075320095347175771972, −5.39436700425051917743986234500, −5.33798780701046543772875311323, −4.70439519283125922258333212217, −4.54198130000904251818904302117, −3.96382480258750722175100470164, −3.74867636289858712945445810088, −3.22154256154143897431808910779, −3.11042504244127140452620438963, −2.52552091026484884433703310692, −2.12105416602992355618330437316, −1.52573970777887798524463564505, −1.25206818586677166841824873256, 0, 0,
1.25206818586677166841824873256, 1.52573970777887798524463564505, 2.12105416602992355618330437316, 2.52552091026484884433703310692, 3.11042504244127140452620438963, 3.22154256154143897431808910779, 3.74867636289858712945445810088, 3.96382480258750722175100470164, 4.54198130000904251818904302117, 4.70439519283125922258333212217, 5.33798780701046543772875311323, 5.39436700425051917743986234500, 6.16126241075320095347175771972, 6.27388854555689088525312454210, 7.04168207828962926948327479357, 7.06468057705941292429778841259, 7.50462914175399962161902402838, 7.70630676768948473668775565710
