L(s) = 1 | − 4·5-s − 3·7-s + 28·11-s − 11·13-s − 44·17-s − 29·19-s + 172·23-s − 109·25-s − 192·29-s − 116·31-s + 12·35-s − 69·37-s − 384·41-s − 328·43-s + 156·47-s − 334·49-s + 392·53-s − 112·55-s + 412·59-s − 425·61-s + 44·65-s − 257·67-s − 1.00e3·71-s − 359·73-s − 84·77-s − 877·79-s − 328·83-s + ⋯ |
L(s) = 1 | − 0.357·5-s − 0.161·7-s + 0.767·11-s − 0.234·13-s − 0.627·17-s − 0.350·19-s + 1.55·23-s − 0.871·25-s − 1.22·29-s − 0.672·31-s + 0.0579·35-s − 0.306·37-s − 1.46·41-s − 1.16·43-s + 0.484·47-s − 0.973·49-s + 1.01·53-s − 0.274·55-s + 0.909·59-s − 0.892·61-s + 0.0839·65-s − 0.468·67-s − 1.67·71-s − 0.575·73-s − 0.124·77-s − 1.24·79-s − 0.433·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 4 T + p^{3} T^{2} \) |
| 7 | \( 1 + 3 T + p^{3} T^{2} \) |
| 11 | \( 1 - 28 T + p^{3} T^{2} \) |
| 13 | \( 1 + 11 T + p^{3} T^{2} \) |
| 17 | \( 1 + 44 T + p^{3} T^{2} \) |
| 19 | \( 1 + 29 T + p^{3} T^{2} \) |
| 23 | \( 1 - 172 T + p^{3} T^{2} \) |
| 29 | \( 1 + 192 T + p^{3} T^{2} \) |
| 31 | \( 1 + 116 T + p^{3} T^{2} \) |
| 37 | \( 1 + 69 T + p^{3} T^{2} \) |
| 41 | \( 1 + 384 T + p^{3} T^{2} \) |
| 43 | \( 1 + 328 T + p^{3} T^{2} \) |
| 47 | \( 1 - 156 T + p^{3} T^{2} \) |
| 53 | \( 1 - 392 T + p^{3} T^{2} \) |
| 59 | \( 1 - 412 T + p^{3} T^{2} \) |
| 61 | \( 1 + 425 T + p^{3} T^{2} \) |
| 67 | \( 1 + 257 T + p^{3} T^{2} \) |
| 71 | \( 1 + 1000 T + p^{3} T^{2} \) |
| 73 | \( 1 + 359 T + p^{3} T^{2} \) |
| 79 | \( 1 + 877 T + p^{3} T^{2} \) |
| 83 | \( 1 + 328 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1572 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1483 T + p^{3} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.31167709528111605645665065820, −9.270913994588705612447507607957, −8.598652058833240682331653994880, −7.37708353601152333423875860401, −6.65938678778736607445910954158, −5.45206378961334940673336548816, −4.29389338683938084714338681265, −3.26180072933711187832071601688, −1.71840369418929046186639320909, 0,
1.71840369418929046186639320909, 3.26180072933711187832071601688, 4.29389338683938084714338681265, 5.45206378961334940673336548816, 6.65938678778736607445910954158, 7.37708353601152333423875860401, 8.598652058833240682331653994880, 9.270913994588705612447507607957, 10.31167709528111605645665065820