| L(s) = 1 | + 0.533·3-s + 1.81·5-s − 3.82·7-s − 2.71·9-s − 0.400·11-s − 1.49·13-s + 0.967·15-s − 2.58·17-s − 2.19·19-s − 2.04·21-s − 4.15·23-s − 1.71·25-s − 3.04·27-s + 7.85·29-s − 4.38·31-s − 0.213·33-s − 6.94·35-s + 3.18·37-s − 0.795·39-s + 10.0·41-s − 6.58·43-s − 4.92·45-s + 1.71·47-s + 7.65·49-s − 1.38·51-s − 0.838·53-s − 0.726·55-s + ⋯ |
| L(s) = 1 | + 0.307·3-s + 0.811·5-s − 1.44·7-s − 0.905·9-s − 0.120·11-s − 0.413·13-s + 0.249·15-s − 0.627·17-s − 0.503·19-s − 0.445·21-s − 0.865·23-s − 0.342·25-s − 0.586·27-s + 1.45·29-s − 0.787·31-s − 0.0372·33-s − 1.17·35-s + 0.523·37-s − 0.127·39-s + 1.57·41-s − 1.00·43-s − 0.734·45-s + 0.250·47-s + 1.09·49-s − 0.193·51-s − 0.115·53-s − 0.0979·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8752 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8752 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.272569154\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.272569154\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 547 | \( 1 + T \) |
| good | 3 | \( 1 - 0.533T + 3T^{2} \) |
| 5 | \( 1 - 1.81T + 5T^{2} \) |
| 7 | \( 1 + 3.82T + 7T^{2} \) |
| 11 | \( 1 + 0.400T + 11T^{2} \) |
| 13 | \( 1 + 1.49T + 13T^{2} \) |
| 17 | \( 1 + 2.58T + 17T^{2} \) |
| 19 | \( 1 + 2.19T + 19T^{2} \) |
| 23 | \( 1 + 4.15T + 23T^{2} \) |
| 29 | \( 1 - 7.85T + 29T^{2} \) |
| 31 | \( 1 + 4.38T + 31T^{2} \) |
| 37 | \( 1 - 3.18T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 + 6.58T + 43T^{2} \) |
| 47 | \( 1 - 1.71T + 47T^{2} \) |
| 53 | \( 1 + 0.838T + 53T^{2} \) |
| 59 | \( 1 - 0.904T + 59T^{2} \) |
| 61 | \( 1 + 2.03T + 61T^{2} \) |
| 67 | \( 1 - 13.7T + 67T^{2} \) |
| 71 | \( 1 + 8.79T + 71T^{2} \) |
| 73 | \( 1 - 14.9T + 73T^{2} \) |
| 79 | \( 1 - 9.78T + 79T^{2} \) |
| 83 | \( 1 - 6.20T + 83T^{2} \) |
| 89 | \( 1 + 12.6T + 89T^{2} \) |
| 97 | \( 1 - 7.34T + 97T^{2} \) |
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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
−7.85540727019103769355173522926, −6.89095321831653671046031269388, −6.28683462433834416820785934985, −5.92335597849755305306899356188, −5.10046895761281312083480240575, −4.12306405768755964219930562103, −3.34392448753926940023738271798, −2.55831993717067800138013897699, −2.08792115997778836382387064456, −0.50049159323987678323194900133,
0.50049159323987678323194900133, 2.08792115997778836382387064456, 2.55831993717067800138013897699, 3.34392448753926940023738271798, 4.12306405768755964219930562103, 5.10046895761281312083480240575, 5.92335597849755305306899356188, 6.28683462433834416820785934985, 6.89095321831653671046031269388, 7.85540727019103769355173522926
