| L(s) = 1 | − 2-s − 3-s + 4-s − 0.467·5-s + 6-s − 4.41·7-s − 8-s + 9-s + 0.467·10-s − 5.53·11-s − 12-s − 5.98·13-s + 4.41·14-s + 0.467·15-s + 16-s − 4.34·17-s − 18-s − 0.467·20-s + 4.41·21-s + 5.53·22-s − 3.16·23-s + 24-s − 4.78·25-s + 5.98·26-s − 27-s − 4.41·28-s + 1.50·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.209·5-s + 0.408·6-s − 1.66·7-s − 0.353·8-s + 0.333·9-s + 0.147·10-s − 1.66·11-s − 0.288·12-s − 1.66·13-s + 1.17·14-s + 0.120·15-s + 0.250·16-s − 1.05·17-s − 0.235·18-s − 0.104·20-s + 0.962·21-s + 1.17·22-s − 0.659·23-s + 0.204·24-s − 0.956·25-s + 1.17·26-s − 0.192·27-s − 0.833·28-s + 0.280·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.07901797350\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07901797350\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 + 0.467T + 5T^{2} \) |
| 7 | \( 1 + 4.41T + 7T^{2} \) |
| 11 | \( 1 + 5.53T + 11T^{2} \) |
| 13 | \( 1 + 5.98T + 13T^{2} \) |
| 17 | \( 1 + 4.34T + 17T^{2} \) |
| 23 | \( 1 + 3.16T + 23T^{2} \) |
| 29 | \( 1 - 1.50T + 29T^{2} \) |
| 31 | \( 1 - 2.18T + 31T^{2} \) |
| 37 | \( 1 - 2.75T + 37T^{2} \) |
| 41 | \( 1 - 1.95T + 41T^{2} \) |
| 43 | \( 1 - 3.59T + 43T^{2} \) |
| 47 | \( 1 + 9.24T + 47T^{2} \) |
| 53 | \( 1 - 2.67T + 53T^{2} \) |
| 59 | \( 1 + 1.32T + 59T^{2} \) |
| 61 | \( 1 + 6.63T + 61T^{2} \) |
| 67 | \( 1 - 2.49T + 67T^{2} \) |
| 71 | \( 1 - 7.59T + 71T^{2} \) |
| 73 | \( 1 - 3.16T + 73T^{2} \) |
| 79 | \( 1 + 1.26T + 79T^{2} \) |
| 83 | \( 1 + 17.9T + 83T^{2} \) |
| 89 | \( 1 + 12.4T + 89T^{2} \) |
| 97 | \( 1 - 8.31T + 97T^{2} \) |
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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
−9.348927594109990296863414918494, −8.187280504515278976660600761167, −7.50921123301525814017976239660, −6.81970406907095281441253216077, −6.08425040842621976481591711199, −5.24141605467512553209939110722, −4.23414418535109839265860834718, −2.92111680654933531969673815537, −2.30239813908795665656746712681, −0.19048034515854953435294516856,
0.19048034515854953435294516856, 2.30239813908795665656746712681, 2.92111680654933531969673815537, 4.23414418535109839265860834718, 5.24141605467512553209939110722, 6.08425040842621976481591711199, 6.81970406907095281441253216077, 7.50921123301525814017976239660, 8.187280504515278976660600761167, 9.348927594109990296863414918494
