L(s) = 1 | − 6·4-s + 2·9-s − 5·16-s + 8·19-s − 72·31-s − 12·36-s + 124·49-s + 328·61-s + 180·64-s − 48·76-s − 552·79-s − 77·81-s − 152·109-s + 444·121-s + 432·124-s + 127-s + 131-s + 137-s + 139-s − 10·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 164·169-s + 16·171-s + ⋯ |
L(s) = 1 | − 3/2·4-s + 2/9·9-s − 0.312·16-s + 8/19·19-s − 2.32·31-s − 1/3·36-s + 2.53·49-s + 5.37·61-s + 2.81·64-s − 0.631·76-s − 6.98·79-s − 0.950·81-s − 1.39·109-s + 3.66·121-s + 3.48·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 0.0694·144-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 0.970·169-s + 0.0935·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8183758871\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8183758871\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{4} T^{4} \) |
| 5 | | \( 1 \) |
good | 2 | $C_2^2$ | \( ( 1 + 3 T^{2} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 62 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 222 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 558 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 878 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 702 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 18 T + p^{2} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 2482 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 558 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 3442 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 1998 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 5598 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 6942 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 82 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 8402 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 5598 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 5182 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 138 T + p^{2} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 4958 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 142 T + p^{2} T^{2} )^{2}( 1 + 142 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 8738 T^{2} + p^{4} T^{4} )^{2} \) |
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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
−10.42312508423581756441210393947, −9.957489967599238255946509188327, −9.918549321675346939345970827289, −9.835576344788790113193440943264, −9.347234988067112550107610244410, −8.840780556043446652023125369066, −8.695916637809971321562132067150, −8.594793722974099512644820849458, −8.465249663615244286862819829368, −7.66099038606770347015452052937, −7.27020219943647043171970581362, −7.24552569594067322114938871160, −6.88725113423520709060004447181, −6.46956883220554981972872824927, −5.69209964193781473695484719826, −5.55406273432819310902086409115, −5.39194695345011453929921597559, −4.87465527927019943797374885879, −4.24628357171964507243442624048, −4.00876422713434947061335680192, −3.96647598190035573314713142173, −3.07297585787477368925610568184, −2.48672149529654840341633430622, −1.76146658001021838191090524086, −0.56232865399743246886736453768,
0.56232865399743246886736453768, 1.76146658001021838191090524086, 2.48672149529654840341633430622, 3.07297585787477368925610568184, 3.96647598190035573314713142173, 4.00876422713434947061335680192, 4.24628357171964507243442624048, 4.87465527927019943797374885879, 5.39194695345011453929921597559, 5.55406273432819310902086409115, 5.69209964193781473695484719826, 6.46956883220554981972872824927, 6.88725113423520709060004447181, 7.24552569594067322114938871160, 7.27020219943647043171970581362, 7.66099038606770347015452052937, 8.465249663615244286862819829368, 8.594793722974099512644820849458, 8.695916637809971321562132067150, 8.840780556043446652023125369066, 9.347234988067112550107610244410, 9.835576344788790113193440943264, 9.918549321675346939345970827289, 9.957489967599238255946509188327, 10.42312508423581756441210393947