| L(s) = 1 | + 4·5-s + 6·9-s − 10·25-s + 20·37-s + 24·45-s − 16·49-s + 16·53-s + 9·81-s − 68·89-s + 28·97-s + 52·113-s − 80·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 12·169-s + 173-s + 179-s + 181-s + 80·185-s + 191-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 2·9-s − 2·25-s + 3.28·37-s + 3.57·45-s − 2.28·49-s + 2.19·53-s + 81-s − 7.20·89-s + 2.84·97-s + 4.89·113-s − 7.15·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.923·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 5.88·185-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.396017775\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.396017775\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2}( 1 + p T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 7 | $C_2^2$ | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 16 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 29 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 64 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 + 29 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 90 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 107 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 5 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 104 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 160 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 17 T + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.58346833560840022942902010968, −6.02646817155231978692560162836, −6.01127146126206876283211027968, −6.00452260883666275322379211693, −5.92129999962561155097613226907, −5.61717676716492713493174662431, −5.27092012538120295092150243821, −5.06744832617590582560101058765, −4.79879379788926519293429243414, −4.60556714529677619921243665335, −4.38179055370844287014344061657, −4.07071741787120852921910169558, −3.93173902905029603075225992029, −3.88113575883600547995745704780, −3.54777590705541205081973212234, −3.08598317769891658658228517315, −2.72851533561087580445039356598, −2.55939272453890653065725893124, −2.47551004291015946916148887625, −1.86922909664973591891713306399, −1.83191393608043881656109789124, −1.48963669398320285573611783192, −1.46076762786418715106388083802, −0.840724524961024415341250550791, −0.41620199703343641056279887952,
0.41620199703343641056279887952, 0.840724524961024415341250550791, 1.46076762786418715106388083802, 1.48963669398320285573611783192, 1.83191393608043881656109789124, 1.86922909664973591891713306399, 2.47551004291015946916148887625, 2.55939272453890653065725893124, 2.72851533561087580445039356598, 3.08598317769891658658228517315, 3.54777590705541205081973212234, 3.88113575883600547995745704780, 3.93173902905029603075225992029, 4.07071741787120852921910169558, 4.38179055370844287014344061657, 4.60556714529677619921243665335, 4.79879379788926519293429243414, 5.06744832617590582560101058765, 5.27092012538120295092150243821, 5.61717676716492713493174662431, 5.92129999962561155097613226907, 6.00452260883666275322379211693, 6.01127146126206876283211027968, 6.02646817155231978692560162836, 6.58346833560840022942902010968
