L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s + 3.08·7-s − 8-s + 9-s − 10-s + 3.95·11-s + 12-s + 3.50·13-s − 3.08·14-s + 15-s + 16-s − 18-s + 3.27·19-s + 20-s + 3.08·21-s − 3.95·22-s − 1.55·23-s − 24-s + 25-s − 3.50·26-s + 27-s + 3.08·28-s + 5.75·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s + 1.16·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s + 1.19·11-s + 0.288·12-s + 0.971·13-s − 0.825·14-s + 0.258·15-s + 0.250·16-s − 0.235·18-s + 0.750·19-s + 0.223·20-s + 0.673·21-s − 0.842·22-s − 0.324·23-s − 0.204·24-s + 0.200·25-s − 0.687·26-s + 0.192·27-s + 0.583·28-s + 1.06·29-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
\(L(1)\) |
\(\approx\) |
\(3.132357326\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.132357326\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.08T + 7T^{2} \) |
| 11 | \( 1 - 3.95T + 11T^{2} \) |
| 13 | \( 1 - 3.50T + 13T^{2} \) |
| 19 | \( 1 - 3.27T + 19T^{2} \) |
| 23 | \( 1 + 1.55T + 23T^{2} \) |
| 29 | \( 1 - 5.75T + 29T^{2} \) |
| 31 | \( 1 + 2.55T + 31T^{2} \) |
| 37 | \( 1 - 5.91T + 37T^{2} \) |
| 41 | \( 1 - 3.64T + 41T^{2} \) |
| 43 | \( 1 - 5.39T + 43T^{2} \) |
| 47 | \( 1 - 2.90T + 47T^{2} \) |
| 53 | \( 1 + 7.75T + 53T^{2} \) |
| 59 | \( 1 + 12.3T + 59T^{2} \) |
| 61 | \( 1 + 2.93T + 61T^{2} \) |
| 67 | \( 1 - 3.87T + 67T^{2} \) |
| 71 | \( 1 + 12.0T + 71T^{2} \) |
| 73 | \( 1 + 10.1T + 73T^{2} \) |
| 79 | \( 1 - 12.8T + 79T^{2} \) |
| 83 | \( 1 - 11.9T + 83T^{2} \) |
| 89 | \( 1 - 1.87T + 89T^{2} \) |
| 97 | \( 1 + 11.2T + 97T^{2} \) |
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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
−7.84330619782990893393814162561, −7.36016966390322880687148993878, −6.35919694668274635896222536591, −5.99711709943305716618563297828, −4.91628437538316682648571343253, −4.20336634943101919336490695041, −3.36736913980473945745419319815, −2.45089030791105123756811272792, −1.50194644921584128734622837312, −1.10190229822733180570345046431,
1.10190229822733180570345046431, 1.50194644921584128734622837312, 2.45089030791105123756811272792, 3.36736913980473945745419319815, 4.20336634943101919336490695041, 4.91628437538316682648571343253, 5.99711709943305716618563297828, 6.35919694668274635896222536591, 7.36016966390322880687148993878, 7.84330619782990893393814162561