| L(s) = 1 | − 2·3-s − 4-s + 6·11-s + 2·12-s − 3·16-s + 12·17-s − 4·19-s + 6·23-s − 7·25-s + 2·27-s − 6·29-s + 2·31-s − 12·33-s − 14·37-s + 10·43-s − 6·44-s − 12·47-s + 6·48-s − 24·51-s − 6·53-s + 8·57-s − 18·59-s + 20·61-s + 7·64-s − 2·67-s − 12·68-s − 12·69-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1/2·4-s + 1.80·11-s + 0.577·12-s − 3/4·16-s + 2.91·17-s − 0.917·19-s + 1.25·23-s − 7/5·25-s + 0.384·27-s − 1.11·29-s + 0.359·31-s − 2.08·33-s − 2.30·37-s + 1.52·43-s − 0.904·44-s − 1.75·47-s + 0.866·48-s − 3.36·51-s − 0.824·53-s + 1.05·57-s − 2.34·59-s + 2.56·61-s + 7/8·64-s − 0.244·67-s − 1.45·68-s − 1.44·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68574961 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68574961 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.683063315\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.683063315\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 12 T + 67 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 6 T + 52 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 2 T + 36 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 10 T + 84 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 12 T + 82 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 6 T + 67 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 18 T + 196 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 20 T + 195 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 2 T + 108 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 123 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 22 T + 252 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 6 T + 148 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 12 T + 166 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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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
−7.990175628779371169469653973427, −7.50940358412757906167455187459, −7.32783132840581600025200395747, −6.81569357477099615136123214417, −6.54929689587369269511445986964, −6.18329660478647655169991671624, −5.91752987531568796858966322458, −5.57100948656734822661607853841, −5.15001605685372899395956367285, −5.07014003367786326064901044551, −4.47812230481827852979817093650, −4.12709760582645491900121620113, −3.64111158332005852101442096013, −3.38470388426645096976732282369, −3.19968034053032770200727283994, −2.32252478225906546509591562030, −1.64844624651709038554093600427, −1.62853272618901808335164087384, −0.72381723488836291932113148643, −0.48943532652671357399362794324,
0.48943532652671357399362794324, 0.72381723488836291932113148643, 1.62853272618901808335164087384, 1.64844624651709038554093600427, 2.32252478225906546509591562030, 3.19968034053032770200727283994, 3.38470388426645096976732282369, 3.64111158332005852101442096013, 4.12709760582645491900121620113, 4.47812230481827852979817093650, 5.07014003367786326064901044551, 5.15001605685372899395956367285, 5.57100948656734822661607853841, 5.91752987531568796858966322458, 6.18329660478647655169991671624, 6.54929689587369269511445986964, 6.81569357477099615136123214417, 7.32783132840581600025200395747, 7.50940358412757906167455187459, 7.990175628779371169469653973427
