L(s) = 1 | + 1.11e3·5-s − 4.37e3·9-s − 5.88e4·17-s + 7.81e5·25-s − 4.89e6·45-s + 9.09e5·49-s − 2.81e6·61-s + 1.30e7·73-s + 1.43e7·81-s − 6.58e7·85-s − 1.99e7·101-s − 3.18e7·121-s + 4.36e8·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 2.57e8·153-s + 157-s + 163-s + 167-s + 1.25e8·169-s + 173-s + 179-s + 181-s + ⋯ |
L(s) = 1 | + 3.99·5-s − 2·9-s − 2.90·17-s + 9.99·25-s − 7.99·45-s + 1.10·49-s − 1.59·61-s + 3.93·73-s + 3·81-s − 11.6·85-s − 1.93·101-s − 1.63·121-s + 19.9·125-s + 5.81·153-s + 2·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(5.206379333\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.206379333\) |
\(L(\frac{9}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + p^{7} T^{2} \) |
good | 3 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 559 T + p^{7} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 1599 T + p^{7} T^{2} )( 1 + 1599 T + p^{7} T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2665 T + p^{7} T^{2} )( 1 + 2665 T + p^{7} T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 29449 T + p^{7} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 15172 T + p^{7} T^{2} )( 1 + 15172 T + p^{7} T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 530963 T + p^{7} T^{2} )( 1 + 530963 T + p^{7} T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 910351 T + p^{7} T^{2} )( 1 + 910351 T + p^{7} T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 1409505 T + p^{7} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6536387 T + p^{7} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 9784528 T + p^{7} T^{2} )( 1 + 9784528 T + p^{7} T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{2} \) |
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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
−10.85847298217529297808156169011, −10.33511802870386048664546893925, −9.517068960395666720450185000107, −9.371266802506446765337132566282, −9.044120223738180870250842043212, −8.681095056124806960498028970085, −8.206611774604628015971274031418, −6.94346740080412907736160564481, −6.56928010675322457610789118313, −6.27114593828088651216423639656, −5.92843800955885736871758506721, −5.24120618715653967771112886692, −5.22778803069231381079456821909, −4.44671945328702501089265430431, −3.19312921543044592579713601447, −2.49173209997785977165324537252, −2.37153125529446514722718933871, −2.00321162215839896372174995657, −1.26768935421771080633769276111, −0.45592302445388498594973455676,
0.45592302445388498594973455676, 1.26768935421771080633769276111, 2.00321162215839896372174995657, 2.37153125529446514722718933871, 2.49173209997785977165324537252, 3.19312921543044592579713601447, 4.44671945328702501089265430431, 5.22778803069231381079456821909, 5.24120618715653967771112886692, 5.92843800955885736871758506721, 6.27114593828088651216423639656, 6.56928010675322457610789118313, 6.94346740080412907736160564481, 8.206611774604628015971274031418, 8.681095056124806960498028970085, 9.044120223738180870250842043212, 9.371266802506446765337132566282, 9.517068960395666720450185000107, 10.33511802870386048664546893925, 10.85847298217529297808156169011