| L(s) = 1 | + 3-s + 8·7-s + 9-s − 12·13-s + 8·19-s + 8·21-s + 25-s + 27-s − 12·37-s − 12·39-s − 8·43-s + 34·49-s + 8·57-s + 12·61-s + 8·63-s − 8·67-s − 28·73-s + 75-s + 32·79-s + 81-s − 96·91-s + 4·97-s + 8·103-s − 20·109-s − 12·111-s − 12·117-s − 22·121-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 3.02·7-s + 1/3·9-s − 3.32·13-s + 1.83·19-s + 1.74·21-s + 1/5·25-s + 0.192·27-s − 1.97·37-s − 1.92·39-s − 1.21·43-s + 34/7·49-s + 1.05·57-s + 1.53·61-s + 1.00·63-s − 0.977·67-s − 3.27·73-s + 0.115·75-s + 3.60·79-s + 1/9·81-s − 10.0·91-s + 0.406·97-s + 0.788·103-s − 1.91·109-s − 1.13·111-s − 1.10·117-s − 2·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.010095125\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.010095125\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−10.34021526003759803966464356447, −9.686400747849369995880150049093, −9.138788409882082860623313179234, −8.609245026512276947697146238421, −7.929044973464791963060046214031, −7.67822247307498044046677545125, −7.34594365176835478558310775458, −6.82423053307536595497404589251, −5.28897888504504330389281711769, −5.12597312940943561354086695880, −4.92167313563440441781925895939, −4.13728859524865050162716547657, −2.97865181251010540219662645668, −2.19180563382997996757520036314, −1.55129913823425505871935815725,
1.55129913823425505871935815725, 2.19180563382997996757520036314, 2.97865181251010540219662645668, 4.13728859524865050162716547657, 4.92167313563440441781925895939, 5.12597312940943561354086695880, 5.28897888504504330389281711769, 6.82423053307536595497404589251, 7.34594365176835478558310775458, 7.67822247307498044046677545125, 7.929044973464791963060046214031, 8.609245026512276947697146238421, 9.138788409882082860623313179234, 9.686400747849369995880150049093, 10.34021526003759803966464356447
