| L(s) = 1 | − 2-s + 0.181·3-s + 4-s + 4.29·5-s − 0.181·6-s − 3.68·7-s − 8-s − 2.96·9-s − 4.29·10-s + 11-s + 0.181·12-s + 3.37·13-s + 3.68·14-s + 0.779·15-s + 16-s − 6.28·17-s + 2.96·18-s + 6.51·19-s + 4.29·20-s − 0.670·21-s − 22-s + 2.48·23-s − 0.181·24-s + 13.4·25-s − 3.37·26-s − 1.08·27-s − 3.68·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.104·3-s + 0.5·4-s + 1.91·5-s − 0.0742·6-s − 1.39·7-s − 0.353·8-s − 0.988·9-s − 1.35·10-s + 0.301·11-s + 0.0524·12-s + 0.936·13-s + 0.985·14-s + 0.201·15-s + 0.250·16-s − 1.52·17-s + 0.699·18-s + 1.49·19-s + 0.959·20-s − 0.146·21-s − 0.213·22-s + 0.517·23-s − 0.0371·24-s + 2.68·25-s − 0.662·26-s − 0.208·27-s − 0.696·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.707369879\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.707369879\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 197 | \( 1 + T \) |
| good | 3 | \( 1 - 0.181T + 3T^{2} \) |
| 5 | \( 1 - 4.29T + 5T^{2} \) |
| 7 | \( 1 + 3.68T + 7T^{2} \) |
| 13 | \( 1 - 3.37T + 13T^{2} \) |
| 17 | \( 1 + 6.28T + 17T^{2} \) |
| 19 | \( 1 - 6.51T + 19T^{2} \) |
| 23 | \( 1 - 2.48T + 23T^{2} \) |
| 29 | \( 1 + 4.57T + 29T^{2} \) |
| 31 | \( 1 - 4.34T + 31T^{2} \) |
| 37 | \( 1 - 6.51T + 37T^{2} \) |
| 41 | \( 1 - 8.42T + 41T^{2} \) |
| 43 | \( 1 - 1.92T + 43T^{2} \) |
| 47 | \( 1 + 11.1T + 47T^{2} \) |
| 53 | \( 1 + 8.28T + 53T^{2} \) |
| 59 | \( 1 - 3.39T + 59T^{2} \) |
| 61 | \( 1 + 9.66T + 61T^{2} \) |
| 67 | \( 1 + 0.980T + 67T^{2} \) |
| 71 | \( 1 + 3.36T + 71T^{2} \) |
| 73 | \( 1 - 7.35T + 73T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 + 17.8T + 83T^{2} \) |
| 89 | \( 1 - 14.9T + 89T^{2} \) |
| 97 | \( 1 - 18.4T + 97T^{2} \) |
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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
−8.876937678435667409491344844342, −7.67231224966737380965624569126, −6.65074903836746745084440248329, −6.19487506832166289055607475008, −5.88336593352172401752740131349, −4.85013369662326339974339605947, −3.34088598811989389298290390634, −2.80320745999024528578388558783, −1.94145812455568794448628820970, −0.811484179273851007849386963483,
0.811484179273851007849386963483, 1.94145812455568794448628820970, 2.80320745999024528578388558783, 3.34088598811989389298290390634, 4.85013369662326339974339605947, 5.88336593352172401752740131349, 6.19487506832166289055607475008, 6.65074903836746745084440248329, 7.67231224966737380965624569126, 8.876937678435667409491344844342
