| L(s) = 1 | − 4·3-s + 16·7-s + 7·9-s − 24·13-s − 12·19-s − 64·21-s + 8·27-s − 68·31-s − 88·37-s + 96·39-s − 56·43-s + 94·49-s + 48·57-s + 148·61-s + 112·63-s − 184·67-s − 112·73-s − 156·79-s − 95·81-s − 384·91-s + 272·93-s + 64·97-s − 208·103-s − 148·109-s + 352·111-s − 168·117-s + 162·121-s + ⋯ |
| L(s) = 1 | − 4/3·3-s + 16/7·7-s + 7/9·9-s − 1.84·13-s − 0.631·19-s − 3.04·21-s + 8/27·27-s − 2.19·31-s − 2.37·37-s + 2.46·39-s − 1.30·43-s + 1.91·49-s + 0.842·57-s + 2.42·61-s + 16/9·63-s − 2.74·67-s − 1.53·73-s − 1.97·79-s − 1.17·81-s − 4.21·91-s + 2.92·93-s + 0.659·97-s − 2.01·103-s − 1.35·109-s + 3.17·111-s − 1.43·117-s + 1.33·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.02879334512\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02879334512\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 + 4 T + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - 8 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 162 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 12 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 402 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 1038 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 962 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 34 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 44 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 3042 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 28 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 4398 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 3998 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 2718 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 74 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 92 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 7202 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 56 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 78 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 3198 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 15522 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 32 T + p^{2} T^{2} )^{2} \) |
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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
−10.12130430776322897074934383530, −9.381148012209776694428432398994, −8.716560787339060164520072761935, −8.626337618457041154734103106579, −8.161225969559709112158463246525, −7.45250895326474080174875393355, −7.36753118731072624475187891384, −6.96901624091407431197888163432, −6.47536714693736669461728929620, −5.67045289740197401628108333306, −5.50057928494852826621864853959, −4.96586557173428742216817035374, −4.93530291730239175372842937399, −4.35973384004040352201615041646, −3.85543903718180553038762067907, −3.01473452754032357707919219411, −2.26907745386494769207279181636, −1.66974236777283868618548711143, −1.42517514556823433301440909801, −0.05394864981299088934945984543,
0.05394864981299088934945984543, 1.42517514556823433301440909801, 1.66974236777283868618548711143, 2.26907745386494769207279181636, 3.01473452754032357707919219411, 3.85543903718180553038762067907, 4.35973384004040352201615041646, 4.93530291730239175372842937399, 4.96586557173428742216817035374, 5.50057928494852826621864853959, 5.67045289740197401628108333306, 6.47536714693736669461728929620, 6.96901624091407431197888163432, 7.36753118731072624475187891384, 7.45250895326474080174875393355, 8.161225969559709112158463246525, 8.626337618457041154734103106579, 8.716560787339060164520072761935, 9.381148012209776694428432398994, 10.12130430776322897074934383530
