| L(s) = 1 | + 3-s + 2·7-s + 9-s + 4·13-s − 17-s + 2·21-s − 5·25-s + 27-s − 2·29-s + 8·31-s − 2·37-s + 4·39-s − 2·41-s − 8·43-s − 6·47-s − 3·49-s − 51-s + 10·53-s + 6·59-s + 6·61-s + 2·63-s − 16·67-s + 2·73-s − 5·75-s + 14·79-s + 81-s − 4·83-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.755·7-s + 1/3·9-s + 1.10·13-s − 0.242·17-s + 0.436·21-s − 25-s + 0.192·27-s − 0.371·29-s + 1.43·31-s − 0.328·37-s + 0.640·39-s − 0.312·41-s − 1.21·43-s − 0.875·47-s − 3/7·49-s − 0.140·51-s + 1.37·53-s + 0.781·59-s + 0.768·61-s + 0.251·63-s − 1.95·67-s + 0.234·73-s − 0.577·75-s + 1.57·79-s + 1/9·81-s − 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.699382800\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.699382800\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 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 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.69114612779044, −13.38674971184431, −13.04261857836256, −12.19997674254911, −11.72207914595914, −11.44474163214044, −10.80148028004421, −10.28594328841857, −9.845280045775597, −9.249654929698366, −8.658002906464233, −8.215571390482252, −8.032289437782058, −7.247333686854413, −6.720760589381024, −6.187443718353170, −5.573782779695039, −4.967156681746610, −4.397577067152476, −3.827415288629189, −3.328428998106007, −2.617480473739803, −1.865708349408896, −1.465709081249036, −0.5825970659752648,
0.5825970659752648, 1.465709081249036, 1.865708349408896, 2.617480473739803, 3.328428998106007, 3.827415288629189, 4.397577067152476, 4.967156681746610, 5.573782779695039, 6.187443718353170, 6.720760589381024, 7.247333686854413, 8.032289437782058, 8.215571390482252, 8.658002906464233, 9.249654929698366, 9.845280045775597, 10.28594328841857, 10.80148028004421, 11.44474163214044, 11.72207914595914, 12.19997674254911, 13.04261857836256, 13.38674971184431, 13.69114612779044
