| L(s) = 1 | − 0.618·2-s + 2.23·3-s − 1.61·4-s − 1.38·6-s − 3.23·7-s + 2.23·8-s + 2.00·9-s − 5.23·11-s − 3.61·12-s − 3·13-s + 2.00·14-s + 1.85·16-s − 0.763·17-s − 1.23·18-s − 2·19-s − 7.23·21-s + 3.23·22-s − 23-s + 5.00·24-s + 1.85·26-s − 2.23·27-s + 5.23·28-s − 3·29-s + 6.70·31-s − 5.61·32-s − 11.7·33-s + 0.472·34-s + ⋯ |
| L(s) = 1 | − 0.437·2-s + 1.29·3-s − 0.809·4-s − 0.564·6-s − 1.22·7-s + 0.790·8-s + 0.666·9-s − 1.57·11-s − 1.04·12-s − 0.832·13-s + 0.534·14-s + 0.463·16-s − 0.185·17-s − 0.291·18-s − 0.458·19-s − 1.57·21-s + 0.689·22-s − 0.208·23-s + 1.02·24-s + 0.363·26-s − 0.430·27-s + 0.989·28-s − 0.557·29-s + 1.20·31-s − 0.993·32-s − 2.03·33-s + 0.0809·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + 0.618T + 2T^{2} \) |
| 3 | \( 1 - 2.23T + 3T^{2} \) |
| 7 | \( 1 + 3.23T + 7T^{2} \) |
| 11 | \( 1 + 5.23T + 11T^{2} \) |
| 13 | \( 1 + 3T + 13T^{2} \) |
| 17 | \( 1 + 0.763T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 - 6.70T + 31T^{2} \) |
| 37 | \( 1 - 1.23T + 37T^{2} \) |
| 41 | \( 1 + 3.47T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 2.23T + 47T^{2} \) |
| 53 | \( 1 + 0.472T + 53T^{2} \) |
| 59 | \( 1 - 6.47T + 59T^{2} \) |
| 61 | \( 1 + 6.94T + 61T^{2} \) |
| 67 | \( 1 - 2.76T + 67T^{2} \) |
| 71 | \( 1 - 12.2T + 71T^{2} \) |
| 73 | \( 1 + 6.52T + 73T^{2} \) |
| 79 | \( 1 + 10.9T + 79T^{2} \) |
| 83 | \( 1 - 8.76T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 + 17.7T + 97T^{2} \) |
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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
−9.867412352734324197405564353661, −9.513513290924372619826046740794, −8.485422600718401530227578063137, −7.962069160367340893428268238224, −7.03378099331315442277711605203, −5.59218035581207484624801786429, −4.42591766911930171458881528185, −3.25780320302808949390923485814, −2.38221053538701941109925317217, 0,
2.38221053538701941109925317217, 3.25780320302808949390923485814, 4.42591766911930171458881528185, 5.59218035581207484624801786429, 7.03378099331315442277711605203, 7.962069160367340893428268238224, 8.485422600718401530227578063137, 9.513513290924372619826046740794, 9.867412352734324197405564353661
