| L(s) = 1 | − 2·3-s − 4-s + 4·7-s + 3·9-s − 8·11-s + 2·12-s + 16-s − 8·21-s − 25-s − 4·27-s − 4·28-s + 16·33-s − 3·36-s − 12·37-s − 4·41-s + 8·44-s − 12·47-s − 2·48-s − 2·49-s − 8·53-s + 12·63-s − 64-s − 24·67-s + 24·71-s − 20·73-s + 2·75-s − 32·77-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1/2·4-s + 1.51·7-s + 9-s − 2.41·11-s + 0.577·12-s + 1/4·16-s − 1.74·21-s − 1/5·25-s − 0.769·27-s − 0.755·28-s + 2.78·33-s − 1/2·36-s − 1.97·37-s − 0.624·41-s + 1.20·44-s − 1.75·47-s − 0.288·48-s − 2/7·49-s − 1.09·53-s + 1.51·63-s − 1/8·64-s − 2.93·67-s + 2.84·71-s − 2.34·73-s + 0.230·75-s − 3.64·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3933903996\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3933903996\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 37 | $C_2$ | \( 1 + 12 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 158 T^{2} + p^{2} T^{4} \) |
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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
−10.25233601242330338090564003971, −9.549344045755351280133870684840, −9.540082499143076009899418847400, −8.606803320091890227767562289421, −8.275500262423017108059551941863, −7.941092806026474074300442167043, −7.82373939850816971140897808123, −6.97706187661899317846342614308, −6.90922549752020937169875704460, −6.08364303326375243717514241185, −5.58482274861421329117222928399, −5.19831875646926909265083324229, −5.10598899958593967280850762312, −4.55093287159720382558945460654, −4.24763845043794295734265975699, −3.23654449668267964420466352171, −2.91158535905479085598150828743, −1.72754324329536463598166408976, −1.70132295377990458461839531936, −0.29542925061326266326070552891,
0.29542925061326266326070552891, 1.70132295377990458461839531936, 1.72754324329536463598166408976, 2.91158535905479085598150828743, 3.23654449668267964420466352171, 4.24763845043794295734265975699, 4.55093287159720382558945460654, 5.10598899958593967280850762312, 5.19831875646926909265083324229, 5.58482274861421329117222928399, 6.08364303326375243717514241185, 6.90922549752020937169875704460, 6.97706187661899317846342614308, 7.82373939850816971140897808123, 7.941092806026474074300442167043, 8.275500262423017108059551941863, 8.606803320091890227767562289421, 9.540082499143076009899418847400, 9.549344045755351280133870684840, 10.25233601242330338090564003971
