| L(s) = 1 | − 12·3-s − 16·5-s + 90·9-s + 192·15-s + 23·16-s − 28·23-s + 142·25-s − 540·27-s + 80·31-s + 28·37-s − 1.44e3·45-s − 172·47-s − 276·48-s + 68·53-s + 88·59-s − 188·67-s + 336·69-s − 1.70e3·75-s − 368·80-s + 2.83e3·81-s + 400·89-s − 960·93-s + 172·97-s + 28·103-s − 336·111-s + 212·113-s + 448·115-s + ⋯ |
| L(s) = 1 | − 4·3-s − 3.19·5-s + 10·9-s + 64/5·15-s + 1.43·16-s − 1.21·23-s + 5.67·25-s − 20·27-s + 2.58·31-s + 0.756·37-s − 32·45-s − 3.65·47-s − 5.75·48-s + 1.28·53-s + 1.49·59-s − 2.80·67-s + 4.86·69-s − 22.7·75-s − 4.59·80-s + 35·81-s + 4.49·89-s − 10.3·93-s + 1.77·97-s + 0.271·103-s − 3.02·111-s + 1.87·113-s + 3.89·115-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1380137496\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1380137496\) |
| \(L(2)\) |
|
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 | $C_1$ | \( ( 1 + p T )^{4} \) |
| 5 | $C_2$ | \( ( 1 + 8 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 82 T^{2} + p^{4} T^{4} \) |
| good | 2 | $C_2^3$ | \( 1 - 23 T^{4} + p^{8} T^{8} \) |
| 7 | $C_2^3$ | \( 1 + 1282 T^{4} + p^{8} T^{8} \) |
| 13 | $C_2^3$ | \( 1 + 44002 T^{4} + p^{8} T^{8} \) |
| 17 | $C_2^3$ | \( 1 - 5438 T^{4} + p^{8} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + 562 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 14 T + 98 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 1322 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 20 T + p^{2} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 14 T + 98 T^{2} - 14 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 1478 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 + 549922 T^{4} + p^{8} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 + 86 T + 3698 T^{2} + 86 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 90 T + p^{2} T^{2} )^{2}( 1 + 56 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 22 T + p^{2} T^{2} )^{4} \) |
| 61 | $C_2^2$ | \( ( 1 + 1558 T^{2} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 94 T + 4418 T^{2} + 94 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 4318 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^3$ | \( 1 + 36867202 T^{4} + p^{8} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 + 12442 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 51909278 T^{4} + p^{8} T^{8} \) |
| 89 | $C_2$ | \( ( 1 - 100 T + p^{2} T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 - 86 T + 3698 T^{2} - 86 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−9.401910680671323001493922887779, −8.708233379118196564152691709577, −8.534092574162266787610122856903, −7.969917895895861624085367761883, −7.914751280008055650758189281050, −7.79277819187836171757978636298, −7.28698936576666739582535603796, −7.27764614460008755274870726654, −6.88018361450821282193794470153, −6.55476089761840595762889188947, −6.10669928088490121067431041171, −6.04724562712841034646163230614, −6.00951771345949351299846562162, −5.16166650956687640814784666138, −5.03858702575652500148514561047, −4.66987124642198007116386307474, −4.62024092154609501269514156275, −4.20711322060649126654161278680, −3.90314218138986698128611897924, −3.44479765125009173358696113453, −3.33633731689427410580707020239, −2.02618575779198540181228204384, −1.13804576693341082361230013129, −0.815873772553838929220468044392, −0.28958016450451157587762534494,
0.28958016450451157587762534494, 0.815873772553838929220468044392, 1.13804576693341082361230013129, 2.02618575779198540181228204384, 3.33633731689427410580707020239, 3.44479765125009173358696113453, 3.90314218138986698128611897924, 4.20711322060649126654161278680, 4.62024092154609501269514156275, 4.66987124642198007116386307474, 5.03858702575652500148514561047, 5.16166650956687640814784666138, 6.00951771345949351299846562162, 6.04724562712841034646163230614, 6.10669928088490121067431041171, 6.55476089761840595762889188947, 6.88018361450821282193794470153, 7.27764614460008755274870726654, 7.28698936576666739582535603796, 7.79277819187836171757978636298, 7.914751280008055650758189281050, 7.969917895895861624085367761883, 8.534092574162266787610122856903, 8.708233379118196564152691709577, 9.401910680671323001493922887779
