L(s) = 1 | − 2-s − 3-s + 4-s + 4.19·5-s + 6-s − 2.69·7-s − 8-s + 9-s − 4.19·10-s + 1.19·11-s − 12-s − 4.58·13-s + 2.69·14-s − 4.19·15-s + 16-s − 17-s − 18-s − 1.48·19-s + 4.19·20-s + 2.69·21-s − 1.19·22-s − 7.59·23-s + 24-s + 12.6·25-s + 4.58·26-s − 27-s − 2.69·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.87·5-s + 0.408·6-s − 1.01·7-s − 0.353·8-s + 0.333·9-s − 1.32·10-s + 0.360·11-s − 0.288·12-s − 1.27·13-s + 0.721·14-s − 1.08·15-s + 0.250·16-s − 0.242·17-s − 0.235·18-s − 0.340·19-s + 0.938·20-s + 0.588·21-s − 0.254·22-s − 1.58·23-s + 0.204·24-s + 2.52·25-s + 0.898·26-s − 0.192·27-s − 0.509·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
good | 5 | \( 1 - 4.19T + 5T^{2} \) |
| 7 | \( 1 + 2.69T + 7T^{2} \) |
| 11 | \( 1 - 1.19T + 11T^{2} \) |
| 13 | \( 1 + 4.58T + 13T^{2} \) |
| 19 | \( 1 + 1.48T + 19T^{2} \) |
| 23 | \( 1 + 7.59T + 23T^{2} \) |
| 29 | \( 1 - 1.17T + 29T^{2} \) |
| 31 | \( 1 - 4.98T + 31T^{2} \) |
| 37 | \( 1 - 0.394T + 37T^{2} \) |
| 41 | \( 1 - 8.27T + 41T^{2} \) |
| 43 | \( 1 + 0.00309T + 43T^{2} \) |
| 47 | \( 1 - 2.16T + 47T^{2} \) |
| 53 | \( 1 + 4.75T + 53T^{2} \) |
| 61 | \( 1 - 3.91T + 61T^{2} \) |
| 67 | \( 1 - 13.6T + 67T^{2} \) |
| 71 | \( 1 + 12.7T + 71T^{2} \) |
| 73 | \( 1 + 2.98T + 73T^{2} \) |
| 79 | \( 1 - 6.59T + 79T^{2} \) |
| 83 | \( 1 + 15.7T + 83T^{2} \) |
| 89 | \( 1 - 2.83T + 89T^{2} \) |
| 97 | \( 1 + 19.2T + 97T^{2} \) |
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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
−7.64177524569760808872556229365, −6.76175016247414115800021494218, −6.35458044355952516674783448231, −5.85715652703644739264472884896, −5.09719634013657599695597159561, −4.12120099192771347363958324315, −2.75243229577905168254852450307, −2.27437897064725808886024733764, −1.28366019176610705564904658565, 0,
1.28366019176610705564904658565, 2.27437897064725808886024733764, 2.75243229577905168254852450307, 4.12120099192771347363958324315, 5.09719634013657599695597159561, 5.85715652703644739264472884896, 6.35458044355952516674783448231, 6.76175016247414115800021494218, 7.64177524569760808872556229365