| L(s) = 1 | − 4-s − 7-s − 2·9-s + 2·13-s − 3·16-s + 6·17-s + 8·19-s + 2·25-s + 28-s + 2·36-s − 8·37-s + 6·41-s + 49-s − 2·52-s + 12·53-s + 2·61-s + 2·63-s + 7·64-s + 16·67-s − 6·68-s − 10·73-s − 8·76-s − 5·81-s + 12·83-s − 2·91-s − 2·100-s + 18·101-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 0.377·7-s − 2/3·9-s + 0.554·13-s − 3/4·16-s + 1.45·17-s + 1.83·19-s + 2/5·25-s + 0.188·28-s + 1/3·36-s − 1.31·37-s + 0.937·41-s + 1/7·49-s − 0.277·52-s + 1.64·53-s + 0.256·61-s + 0.251·63-s + 7/8·64-s + 1.95·67-s − 0.727·68-s − 1.17·73-s − 0.917·76-s − 5/9·81-s + 1.31·83-s − 0.209·91-s − 1/5·100-s + 1.79·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41503 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41503 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.114337095\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.114337095\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 11 | $C_2$ | \( 1 + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
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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
−10.08413165334841815330474659559, −9.749659497981391797203883443975, −9.234516000430137164523456704711, −8.667044376361886927783253520503, −8.365969422532131255143167817892, −7.46527880275731058373331702716, −7.26928094611484628076589803252, −6.43906336777089411657390297016, −5.75141901363088440822485038831, −5.35388325492838072186666256000, −4.77442524762674841784226821531, −3.71668907115242125030065328458, −3.38112485638183544262061635605, −2.47131910765623514659467569057, −1.00416546591001491474959424526,
1.00416546591001491474959424526, 2.47131910765623514659467569057, 3.38112485638183544262061635605, 3.71668907115242125030065328458, 4.77442524762674841784226821531, 5.35388325492838072186666256000, 5.75141901363088440822485038831, 6.43906336777089411657390297016, 7.26928094611484628076589803252, 7.46527880275731058373331702716, 8.365969422532131255143167817892, 8.667044376361886927783253520503, 9.234516000430137164523456704711, 9.749659497981391797203883443975, 10.08413165334841815330474659559
