| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 7-s + 8-s − 2·9-s − 10-s + 12-s − 7·13-s + 14-s − 15-s + 16-s − 6·17-s − 2·18-s + 5·19-s − 20-s + 21-s + 9·23-s + 24-s + 25-s − 7·26-s − 5·27-s + 28-s − 30-s + 2·31-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s − 2/3·9-s − 0.316·10-s + 0.288·12-s − 1.94·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s − 1.45·17-s − 0.471·18-s + 1.14·19-s − 0.223·20-s + 0.218·21-s + 1.87·23-s + 0.204·24-s + 1/5·25-s − 1.37·26-s − 0.962·27-s + 0.188·28-s − 0.182·30-s + 0.359·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.195611372\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.195611372\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−7.59900369916854803267678751337, −7.22041862310026655693171304037, −6.50719547244182618761498756486, −5.43038104181122990998275902632, −4.95900322062968505408506954682, −4.35557102889788249713789067340, −3.41223047941465300615735975884, −2.63245063310640201828184191296, −2.26355199478866896280864715948, −0.73342534152403238635157286666,
0.73342534152403238635157286666, 2.26355199478866896280864715948, 2.63245063310640201828184191296, 3.41223047941465300615735975884, 4.35557102889788249713789067340, 4.95900322062968505408506954682, 5.43038104181122990998275902632, 6.50719547244182618761498756486, 7.22041862310026655693171304037, 7.59900369916854803267678751337
