| L(s) = 1 | − 5-s − 2·7-s + 2·11-s + 2·13-s − 4·17-s + 25-s − 2·29-s − 8·31-s + 2·35-s − 8·37-s + 8·41-s + 43-s − 8·47-s − 3·49-s + 14·53-s − 2·55-s + 10·59-s + 12·61-s − 2·65-s + 12·67-s − 8·71-s − 2·73-s − 4·77-s + 4·79-s + 6·83-s + 4·85-s + 18·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.755·7-s + 0.603·11-s + 0.554·13-s − 0.970·17-s + 1/5·25-s − 0.371·29-s − 1.43·31-s + 0.338·35-s − 1.31·37-s + 1.24·41-s + 0.152·43-s − 1.16·47-s − 3/7·49-s + 1.92·53-s − 0.269·55-s + 1.30·59-s + 1.53·61-s − 0.248·65-s + 1.46·67-s − 0.949·71-s − 0.234·73-s − 0.455·77-s + 0.450·79-s + 0.658·83-s + 0.433·85-s + 1.90·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.327009480\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.327009480\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−13.28656872614530, −13.03890279239497, −12.79485418066313, −11.91720776188466, −11.69717876543462, −11.15208222165578, −10.64059760004690, −10.23443437771138, −9.418271113298571, −9.265616587214665, −8.565925306120462, −8.341984132084862, −7.373795781669610, −7.180364407071111, −6.472339023637079, −6.242080628964981, −5.392952607580212, −5.040892032620218, −4.126768433080684, −3.733504976620848, −3.461503803060988, −2.491263854526830, −2.022848223194191, −1.143925588548737, −0.3765342065980999,
0.3765342065980999, 1.143925588548737, 2.022848223194191, 2.491263854526830, 3.461503803060988, 3.733504976620848, 4.126768433080684, 5.040892032620218, 5.392952607580212, 6.242080628964981, 6.472339023637079, 7.180364407071111, 7.373795781669610, 8.341984132084862, 8.565925306120462, 9.265616587214665, 9.418271113298571, 10.23443437771138, 10.64059760004690, 11.15208222165578, 11.69717876543462, 11.91720776188466, 12.79485418066313, 13.03890279239497, 13.28656872614530
