| L(s) = 1 | + 4·5-s + 2·7-s − 9-s − 6·13-s + 11·25-s − 8·29-s + 8·35-s + 22·37-s − 4·45-s + 24·47-s − 11·49-s − 14·61-s − 2·63-s − 24·65-s + 8·67-s + 4·73-s − 30·79-s + 81-s + 24·83-s − 12·91-s − 34·97-s + 24·101-s + 6·117-s + 13·121-s + 24·125-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 0.755·7-s − 1/3·9-s − 1.66·13-s + 11/5·25-s − 1.48·29-s + 1.35·35-s + 3.61·37-s − 0.596·45-s + 3.50·47-s − 1.57·49-s − 1.79·61-s − 0.251·63-s − 2.97·65-s + 0.977·67-s + 0.468·73-s − 3.37·79-s + 1/9·81-s + 2.63·83-s − 1.25·91-s − 3.45·97-s + 2.38·101-s + 0.554·117-s + 1.18·121-s + 2.14·125-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2433600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2433600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.324611086\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.324611086\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 61 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 153 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 17 T + p T^{2} )^{2} \) |
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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
−9.651726448827967145579778130900, −9.486500492605150895571738300279, −9.039218183412948233651456050231, −8.394366524924111887481354310288, −8.122894790179946397418937552585, −7.51509719093346787509970298428, −7.28329720702231434953923750868, −6.93355418746771645942303069558, −6.09186508455048539771996623855, −5.90896905769907866820549202589, −5.74929684008790663202942341312, −5.09416424934153052338738951143, −4.66145786075988434759631784653, −4.44363754570276828603137786124, −3.68857639034226891738710381903, −2.70792545993724862908562685964, −2.66326682864460851017548196859, −2.09177213827671168142992989833, −1.54540637112691210400902024149, −0.71215689873199042779153107483,
0.71215689873199042779153107483, 1.54540637112691210400902024149, 2.09177213827671168142992989833, 2.66326682864460851017548196859, 2.70792545993724862908562685964, 3.68857639034226891738710381903, 4.44363754570276828603137786124, 4.66145786075988434759631784653, 5.09416424934153052338738951143, 5.74929684008790663202942341312, 5.90896905769907866820549202589, 6.09186508455048539771996623855, 6.93355418746771645942303069558, 7.28329720702231434953923750868, 7.51509719093346787509970298428, 8.122894790179946397418937552585, 8.394366524924111887481354310288, 9.039218183412948233651456050231, 9.486500492605150895571738300279, 9.651726448827967145579778130900
