| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 0.347·7-s − 8-s + 9-s − 10-s + 1.18·11-s − 12-s + 1.53·13-s + 0.347·14-s − 15-s + 16-s − 18-s − 3.69·19-s + 20-s + 0.347·21-s − 1.18·22-s − 2.22·23-s + 24-s + 25-s − 1.53·26-s − 27-s − 0.347·28-s + 4.45·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s − 0.131·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s + 0.357·11-s − 0.288·12-s + 0.424·13-s + 0.0928·14-s − 0.258·15-s + 0.250·16-s − 0.235·18-s − 0.847·19-s + 0.223·20-s + 0.0757·21-s − 0.252·22-s − 0.464·23-s + 0.204·24-s + 0.200·25-s − 0.300·26-s − 0.192·27-s − 0.0656·28-s + 0.826·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.156442768\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.156442768\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 + 0.347T + 7T^{2} \) |
| 11 | \( 1 - 1.18T + 11T^{2} \) |
| 13 | \( 1 - 1.53T + 13T^{2} \) |
| 19 | \( 1 + 3.69T + 19T^{2} \) |
| 23 | \( 1 + 2.22T + 23T^{2} \) |
| 29 | \( 1 - 4.45T + 29T^{2} \) |
| 31 | \( 1 + 3.75T + 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 + 9.82T + 41T^{2} \) |
| 43 | \( 1 + 3.06T + 43T^{2} \) |
| 47 | \( 1 - 1.32T + 47T^{2} \) |
| 53 | \( 1 - 7.18T + 53T^{2} \) |
| 59 | \( 1 - 3.55T + 59T^{2} \) |
| 61 | \( 1 - 3.06T + 61T^{2} \) |
| 67 | \( 1 + 2.24T + 67T^{2} \) |
| 71 | \( 1 - 2.36T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 - 13.1T + 79T^{2} \) |
| 83 | \( 1 + 0.822T + 83T^{2} \) |
| 89 | \( 1 - 3.41T + 89T^{2} \) |
| 97 | \( 1 - 4.32T + 97T^{2} \) |
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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
−7.906162870808779638832510242003, −6.88892205064653648118514321292, −6.50531920787860132178361473292, −5.90494347032343286906878065777, −5.13307610401543868676660767383, −4.26800785657726169489285251934, −3.44035927833408591492089266936, −2.38645290670542643304378577258, −1.60261464468722231467772185660, −0.61568032907891170765511168195,
0.61568032907891170765511168195, 1.60261464468722231467772185660, 2.38645290670542643304378577258, 3.44035927833408591492089266936, 4.26800785657726169489285251934, 5.13307610401543868676660767383, 5.90494347032343286906878065777, 6.50531920787860132178361473292, 6.88892205064653648118514321292, 7.906162870808779638832510242003
