L(s) = 1 | + 2-s − 3-s + 4-s + 0.744·5-s − 6-s − 0.530·7-s + 8-s + 9-s + 0.744·10-s − 1.93·11-s − 12-s − 3.52·13-s − 0.530·14-s − 0.744·15-s + 16-s − 0.585·17-s + 18-s + 19-s + 0.744·20-s + 0.530·21-s − 1.93·22-s + 6.08·23-s − 24-s − 4.44·25-s − 3.52·26-s − 27-s − 0.530·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.332·5-s − 0.408·6-s − 0.200·7-s + 0.353·8-s + 0.333·9-s + 0.235·10-s − 0.582·11-s − 0.288·12-s − 0.978·13-s − 0.141·14-s − 0.192·15-s + 0.250·16-s − 0.141·17-s + 0.235·18-s + 0.229·19-s + 0.166·20-s + 0.115·21-s − 0.411·22-s + 1.26·23-s − 0.204·24-s − 0.889·25-s − 0.691·26-s − 0.192·27-s − 0.100·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 53 | \( 1 + T \) |
good | 5 | \( 1 - 0.744T + 5T^{2} \) |
| 7 | \( 1 + 0.530T + 7T^{2} \) |
| 11 | \( 1 + 1.93T + 11T^{2} \) |
| 13 | \( 1 + 3.52T + 13T^{2} \) |
| 17 | \( 1 + 0.585T + 17T^{2} \) |
| 23 | \( 1 - 6.08T + 23T^{2} \) |
| 29 | \( 1 - 8.24T + 29T^{2} \) |
| 31 | \( 1 + 10.6T + 31T^{2} \) |
| 37 | \( 1 + 3.64T + 37T^{2} \) |
| 41 | \( 1 - 11.3T + 41T^{2} \) |
| 43 | \( 1 + 3.08T + 43T^{2} \) |
| 47 | \( 1 + 0.785T + 47T^{2} \) |
| 59 | \( 1 + 8.73T + 59T^{2} \) |
| 61 | \( 1 + 13.5T + 61T^{2} \) |
| 67 | \( 1 - 7.13T + 67T^{2} \) |
| 71 | \( 1 - 13.3T + 71T^{2} \) |
| 73 | \( 1 + 13.4T + 73T^{2} \) |
| 79 | \( 1 - 15.9T + 79T^{2} \) |
| 83 | \( 1 + 12.4T + 83T^{2} \) |
| 89 | \( 1 + 14.1T + 89T^{2} \) |
| 97 | \( 1 + 11.0T + 97T^{2} \) |
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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
−7.46351122976386371119479026681, −6.93501383593353251318126643017, −6.18598300944663404186163810446, −5.44352225600754241151264118008, −4.97549552736776235476469860929, −4.23323479340560202450401893405, −3.18785707470660497751574789458, −2.48137724712774803097346858956, −1.44309448470157258780314808659, 0,
1.44309448470157258780314808659, 2.48137724712774803097346858956, 3.18785707470660497751574789458, 4.23323479340560202450401893405, 4.97549552736776235476469860929, 5.44352225600754241151264118008, 6.18598300944663404186163810446, 6.93501383593353251318126643017, 7.46351122976386371119479026681