| L(s) = 1 | − 16·7-s + 16·13-s + 64·19-s + 48·25-s − 80·31-s + 52·37-s + 32·43-s + 94·49-s + 108·61-s − 160·67-s + 192·73-s + 208·79-s − 256·91-s − 160·97-s + 144·103-s + 176·109-s + 114·121-s + 127-s + 131-s − 1.02e3·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 2.28·7-s + 1.23·13-s + 3.36·19-s + 1.91·25-s − 2.58·31-s + 1.40·37-s + 0.744·43-s + 1.91·49-s + 1.77·61-s − 2.38·67-s + 2.63·73-s + 2.63·79-s − 2.81·91-s − 1.64·97-s + 1.39·103-s + 1.61·109-s + 0.942·121-s + 0.00787·127-s + 0.00763·131-s − 7.69·133-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.118963653\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.118963653\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 - 48 T^{2} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 8 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 114 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 8 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 416 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 32 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 94 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 240 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 40 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 26 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 1056 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 16 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 4290 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 4560 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6450 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 54 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 80 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 3810 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 96 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 104 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 3410 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 9792 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 80 T + p^{2} T^{2} )^{2} \) |
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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
−10.77606732635183931991648051948, −10.18740867922241108472127756552, −9.643956578120976376456373787195, −9.540442134613457154532251798078, −9.084701781411869464813862659221, −8.829658673050180257031899691127, −7.966221066749094383374244199874, −7.55795371199955245316332923015, −6.98568704424141534151736360160, −6.83954241802546501518369605901, −6.09761832254823018493161628323, −5.82273014353618091713079678807, −5.30119149549375177619160604510, −4.74765162343635167813083583267, −3.60420303233784191357032967189, −3.54396501780259619779561294411, −3.15887468577234989542557758485, −2.43863138338158499598562987458, −1.18023741532820310309672043739, −0.65309059300909970336627435040,
0.65309059300909970336627435040, 1.18023741532820310309672043739, 2.43863138338158499598562987458, 3.15887468577234989542557758485, 3.54396501780259619779561294411, 3.60420303233784191357032967189, 4.74765162343635167813083583267, 5.30119149549375177619160604510, 5.82273014353618091713079678807, 6.09761832254823018493161628323, 6.83954241802546501518369605901, 6.98568704424141534151736360160, 7.55795371199955245316332923015, 7.966221066749094383374244199874, 8.829658673050180257031899691127, 9.084701781411869464813862659221, 9.540442134613457154532251798078, 9.643956578120976376456373787195, 10.18740867922241108472127756552, 10.77606732635183931991648051948
