L(s) = 1 | − 2-s + 3-s + 4-s − 3.41·5-s − 6-s − 7-s − 8-s + 9-s + 3.41·10-s − 3.24·11-s + 12-s + 14-s − 3.41·15-s + 16-s + 1.80·17-s − 18-s − 1.93·19-s − 3.41·20-s − 21-s + 3.24·22-s + 3.28·23-s − 24-s + 6.66·25-s + 27-s − 28-s − 3.64·29-s + 3.41·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.52·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s + 1.07·10-s − 0.979·11-s + 0.288·12-s + 0.267·14-s − 0.881·15-s + 0.250·16-s + 0.437·17-s − 0.235·18-s − 0.444·19-s − 0.763·20-s − 0.218·21-s + 0.692·22-s + 0.684·23-s − 0.204·24-s + 1.33·25-s + 0.192·27-s − 0.188·28-s − 0.676·29-s + 0.623·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 + 3.41T + 5T^{2} \) |
| 11 | \( 1 + 3.24T + 11T^{2} \) |
| 17 | \( 1 - 1.80T + 17T^{2} \) |
| 19 | \( 1 + 1.93T + 19T^{2} \) |
| 23 | \( 1 - 3.28T + 23T^{2} \) |
| 29 | \( 1 + 3.64T + 29T^{2} \) |
| 31 | \( 1 - 3.10T + 31T^{2} \) |
| 37 | \( 1 - 4.80T + 37T^{2} \) |
| 41 | \( 1 - 4.58T + 41T^{2} \) |
| 43 | \( 1 - 5.11T + 43T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 + 2.46T + 53T^{2} \) |
| 59 | \( 1 + 13.2T + 59T^{2} \) |
| 61 | \( 1 + 1.76T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 - 2.66T + 71T^{2} \) |
| 73 | \( 1 + 0.368T + 73T^{2} \) |
| 79 | \( 1 + 15.0T + 79T^{2} \) |
| 83 | \( 1 - 1.43T + 83T^{2} \) |
| 89 | \( 1 + 5.10T + 89T^{2} \) |
| 97 | \( 1 - 11.9T + 97T^{2} \) |
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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
−7.62934484036565802562892396678, −7.34727254815817448897412980366, −6.43080282619867355097342929159, −5.51862235460595579333180261737, −4.51242583228203543330532934938, −3.85558359062283771052041389534, −3.02624309303735036376971597378, −2.44620484750367778742196299500, −1.02631367090756010867941940444, 0,
1.02631367090756010867941940444, 2.44620484750367778742196299500, 3.02624309303735036376971597378, 3.85558359062283771052041389534, 4.51242583228203543330532934938, 5.51862235460595579333180261737, 6.43080282619867355097342929159, 7.34727254815817448897412980366, 7.62934484036565802562892396678