| L(s) = 1 | − 3-s − 3·4-s + 2·5-s + 9-s − 8·11-s + 3·12-s − 4·13-s − 2·15-s + 5·16-s + 4·17-s − 6·20-s + 3·25-s − 27-s + 8·33-s − 3·36-s + 4·39-s + 24·44-s + 2·45-s − 5·48-s − 14·49-s − 4·51-s + 12·52-s − 20·53-s − 16·55-s + 6·60-s − 3·64-s − 8·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 3/2·4-s + 0.894·5-s + 1/3·9-s − 2.41·11-s + 0.866·12-s − 1.10·13-s − 0.516·15-s + 5/4·16-s + 0.970·17-s − 1.34·20-s + 3/5·25-s − 0.192·27-s + 1.39·33-s − 1/2·36-s + 0.640·39-s + 3.61·44-s + 0.298·45-s − 0.721·48-s − 2·49-s − 0.560·51-s + 1.66·52-s − 2.74·53-s − 2.15·55-s + 0.774·60-s − 3/8·64-s − 0.992·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 357075 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 357075 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4723191791\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4723191791\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( 1 + T \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 23 | $C_2$ | \( 1 + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 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
−8.747331681916898422959978426521, −8.094760108755191183963901439773, −7.83112305999270717685638671290, −7.66488013441745380243230842523, −6.71127590095603787571070148787, −6.32926620937961354425038023206, −5.58960569660388488966925360413, −5.23920392624592057055772361749, −4.94550099004770897084787901423, −4.75255696513902627948971988421, −3.81743944541262294264211328806, −3.07668249599801333807028832473, −2.55924503262722823844795003043, −1.65969933567244721093125535055, −0.40136380903884576994501386816,
0.40136380903884576994501386816, 1.65969933567244721093125535055, 2.55924503262722823844795003043, 3.07668249599801333807028832473, 3.81743944541262294264211328806, 4.75255696513902627948971988421, 4.94550099004770897084787901423, 5.23920392624592057055772361749, 5.58960569660388488966925360413, 6.32926620937961354425038023206, 6.71127590095603787571070148787, 7.66488013441745380243230842523, 7.83112305999270717685638671290, 8.094760108755191183963901439773, 8.747331681916898422959978426521
