| L(s) = 1 | − 32·11-s − 48·17-s + 48·19-s + 34·25-s + 16·41-s − 112·43-s + 82·49-s + 64·59-s + 160·67-s + 132·73-s − 32·83-s − 288·89-s + 188·97-s + 192·107-s + 160·113-s + 526·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 334·169-s + 173-s + ⋯ |
| L(s) = 1 | − 2.90·11-s − 2.82·17-s + 2.52·19-s + 1.35·25-s + 0.390·41-s − 2.60·43-s + 1.67·49-s + 1.08·59-s + 2.38·67-s + 1.80·73-s − 0.385·83-s − 3.23·89-s + 1.93·97-s + 1.79·107-s + 1.41·113-s + 4.34·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 1.97·169-s + 0.00578·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.731655276\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.731655276\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 34 T^{2} + p^{4} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 82 T^{2} + p^{4} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 16 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 334 T^{2} + p^{4} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 24 T + p^{2} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 24 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 34 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 254 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 782 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2414 T^{2} + p^{4} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 56 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 3394 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 4322 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 32 T + p^{2} T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 3598 T^{2} + p^{4} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 80 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 6302 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 66 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 12082 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 16 T + p^{2} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 144 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 94 T + p^{2} T^{2} )^{2} \) |
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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
−8.759511770890282911929815815448, −8.634139466132426938120027236310, −8.376135364907829871833171529786, −7.82370628990227040201864058061, −7.44193098919170421722713360005, −7.19990329581567557492871508024, −6.60939513076052826859698496447, −6.58643701833148806334060383167, −5.57310933594971628906782703117, −5.47624942912492165009026177904, −5.01876665500674753136083694001, −4.86353854026262915077933327359, −4.28898652742266672214473141129, −3.66105943761303117820425501390, −2.94078720553002753792694425607, −2.92315899379041569433953463895, −2.21425204502679104608040221773, −1.95730085980960235179380175873, −0.818109151189451698056032792845, −0.41041227621845883016139404210,
0.41041227621845883016139404210, 0.818109151189451698056032792845, 1.95730085980960235179380175873, 2.21425204502679104608040221773, 2.92315899379041569433953463895, 2.94078720553002753792694425607, 3.66105943761303117820425501390, 4.28898652742266672214473141129, 4.86353854026262915077933327359, 5.01876665500674753136083694001, 5.47624942912492165009026177904, 5.57310933594971628906782703117, 6.58643701833148806334060383167, 6.60939513076052826859698496447, 7.19990329581567557492871508024, 7.44193098919170421722713360005, 7.82370628990227040201864058061, 8.376135364907829871833171529786, 8.634139466132426938120027236310, 8.759511770890282911929815815448
