| L(s) = 1 | − 4·7-s − 8·13-s − 22·19-s + 38·25-s − 64·31-s − 68·37-s + 122·43-s − 86·49-s + 112·61-s + 62·67-s + 130·73-s − 76·79-s + 32·91-s − 230·97-s + 80·103-s − 104·109-s + 239·121-s + 127-s + 131-s + 88·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 4/7·7-s − 0.615·13-s − 1.15·19-s + 1.51·25-s − 2.06·31-s − 1.83·37-s + 2.83·43-s − 1.75·49-s + 1.83·61-s + 0.925·67-s + 1.78·73-s − 0.962·79-s + 0.351·91-s − 2.37·97-s + 0.776·103-s − 0.954·109-s + 1.97·121-s + 0.00787·127-s + 0.00763·131-s + 0.661·133-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.003760308\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.003760308\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 - 38 T^{2} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 239 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 335 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 11 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 290 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 346 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 32 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 34 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 3215 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 61 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 2066 T^{2} + p^{4} T^{4} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 4439 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 56 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 31 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 9110 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 65 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 38 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 11426 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 290 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 115 T + p^{2} T^{2} )^{2} \) |
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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
−9.546640761844739276704694706072, −9.411671771052165192746800276857, −8.815546951397439759505793585471, −8.461288725469633810835322432210, −8.240762066469506980399092864943, −7.45515078527246973222836518300, −7.15598319982150878316741748703, −6.91474511622387263210860611461, −6.40491751688745312791164680007, −5.96560014138101375008161867647, −5.39772286030385734806127657254, −5.10156127694762245778807808530, −4.59159809978453365461752437236, −3.93091865301527252882053056880, −3.65617619787777500507816703106, −3.03484075587231055403847347428, −2.41007357527634081730941328510, −2.02108164686237236106965675959, −1.17543726318088450328480598535, −0.30489943536045386666081858761,
0.30489943536045386666081858761, 1.17543726318088450328480598535, 2.02108164686237236106965675959, 2.41007357527634081730941328510, 3.03484075587231055403847347428, 3.65617619787777500507816703106, 3.93091865301527252882053056880, 4.59159809978453365461752437236, 5.10156127694762245778807808530, 5.39772286030385734806127657254, 5.96560014138101375008161867647, 6.40491751688745312791164680007, 6.91474511622387263210860611461, 7.15598319982150878316741748703, 7.45515078527246973222836518300, 8.240762066469506980399092864943, 8.461288725469633810835322432210, 8.815546951397439759505793585471, 9.411671771052165192746800276857, 9.546640761844739276704694706072
