| L(s) = 1 | + 10·9-s + 12·17-s − 36·41-s − 4·49-s + 28·73-s + 57·81-s − 12·89-s + 56·97-s − 60·113-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 120·153-s + 157-s + 163-s + 167-s + 28·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | + 10/3·9-s + 2.91·17-s − 5.62·41-s − 4/7·49-s + 3.27·73-s + 19/3·81-s − 1.27·89-s + 5.68·97-s − 5.64·113-s + 2.36·121-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 + 9.70·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.15·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.763601220\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.763601220\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 37 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 59 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{4} \) |
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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
−6.68226693209946575585033866919, −6.65613428916529316904308832976, −6.41551232282068268546429082149, −6.20768381771529631096708504923, −5.76206524696414855628287636074, −5.49661617337831712320718593387, −5.35140195699131014234739318156, −5.09102860457684433509625386889, −5.08884856980941655817867113235, −4.64932629008631575130717058640, −4.60926622573821709970303468403, −4.28888829682471120431544748555, −3.99981201727396322075655547481, −3.58047722835624578966981436010, −3.56431351919034813383691231886, −3.49546605153785073194898558567, −3.06614619778292456830886088170, −3.02017442303665297041740427907, −2.25146763004207284532689741462, −1.95877629753816336500375151585, −1.87372056865281661912986566625, −1.48570495003332053803400892849, −1.31126170726735790853787954819, −0.922714657948824696762264671392, −0.50984259156794974918424587969,
0.50984259156794974918424587969, 0.922714657948824696762264671392, 1.31126170726735790853787954819, 1.48570495003332053803400892849, 1.87372056865281661912986566625, 1.95877629753816336500375151585, 2.25146763004207284532689741462, 3.02017442303665297041740427907, 3.06614619778292456830886088170, 3.49546605153785073194898558567, 3.56431351919034813383691231886, 3.58047722835624578966981436010, 3.99981201727396322075655547481, 4.28888829682471120431544748555, 4.60926622573821709970303468403, 4.64932629008631575130717058640, 5.08884856980941655817867113235, 5.09102860457684433509625386889, 5.35140195699131014234739318156, 5.49661617337831712320718593387, 5.76206524696414855628287636074, 6.20768381771529631096708504923, 6.41551232282068268546429082149, 6.65613428916529316904308832976, 6.68226693209946575585033866919
