| L(s) = 1 | + 4·3-s − 2·5-s − 4·7-s + 8·9-s + 4·11-s + 4·13-s − 8·15-s − 4·17-s + 2·19-s − 16·21-s − 12·23-s + 3·25-s + 12·27-s − 4·29-s − 8·31-s + 16·33-s + 8·35-s + 12·37-s + 16·39-s − 4·41-s − 4·43-s − 16·45-s − 4·47-s + 6·49-s − 16·51-s − 4·53-s − 8·55-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 0.894·5-s − 1.51·7-s + 8/3·9-s + 1.20·11-s + 1.10·13-s − 2.06·15-s − 0.970·17-s + 0.458·19-s − 3.49·21-s − 2.50·23-s + 3/5·25-s + 2.30·27-s − 0.742·29-s − 1.43·31-s + 2.78·33-s + 1.35·35-s + 1.97·37-s + 2.56·39-s − 0.624·41-s − 0.609·43-s − 2.38·45-s − 0.583·47-s + 6/7·49-s − 2.24·51-s − 0.549·53-s − 1.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36966400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36966400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.657353320\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.657353320\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_4$ | \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 4 T - 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 12 T + 92 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 82 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 90 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 60 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 16 T + 154 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 136 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 16 T + 198 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 142 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 4 T + 110 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 12 T + 212 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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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
−8.178714704335085695066612810443, −8.081841763724723666841258246528, −7.72626455796562468528070299048, −7.31664768757480519744868031083, −6.79376646119022485963412203909, −6.63233322792196481221653788695, −6.20991445565955612462289251183, −6.07480784272957455272522279868, −5.33862443065777222560859966016, −4.89471618834332066390808629513, −4.07386374163690947017807827177, −4.01473351080345241759639070304, −3.69771457842598410922631513083, −3.65043175327109938165024711752, −2.94786755752464223868595662193, −2.82313182011037335449846617568, −2.11789213023505703198082474778, −1.85293454380981843244540192559, −1.22809768222672532943618612491, −0.34469161786079647606581774795,
0.34469161786079647606581774795, 1.22809768222672532943618612491, 1.85293454380981843244540192559, 2.11789213023505703198082474778, 2.82313182011037335449846617568, 2.94786755752464223868595662193, 3.65043175327109938165024711752, 3.69771457842598410922631513083, 4.01473351080345241759639070304, 4.07386374163690947017807827177, 4.89471618834332066390808629513, 5.33862443065777222560859966016, 6.07480784272957455272522279868, 6.20991445565955612462289251183, 6.63233322792196481221653788695, 6.79376646119022485963412203909, 7.31664768757480519744868031083, 7.72626455796562468528070299048, 8.081841763724723666841258246528, 8.178714704335085695066612810443
