| L(s) = 1 | − 1.68·2-s + 3-s + 0.841·4-s − 5-s − 1.68·6-s + 1.95·8-s + 9-s + 1.68·10-s + 1.77·11-s + 0.841·12-s − 3.05·13-s − 15-s − 4.97·16-s + 7.59·17-s − 1.68·18-s − 3.12·19-s − 0.841·20-s − 2.99·22-s − 4.18·23-s + 1.95·24-s + 25-s + 5.14·26-s + 27-s + 7.08·29-s + 1.68·30-s − 0.871·31-s + 4.47·32-s + ⋯ |
| L(s) = 1 | − 1.19·2-s + 0.577·3-s + 0.420·4-s − 0.447·5-s − 0.688·6-s + 0.690·8-s + 0.333·9-s + 0.533·10-s + 0.535·11-s + 0.242·12-s − 0.846·13-s − 0.258·15-s − 1.24·16-s + 1.84·17-s − 0.397·18-s − 0.717·19-s − 0.188·20-s − 0.637·22-s − 0.871·23-s + 0.398·24-s + 0.200·25-s + 1.00·26-s + 0.192·27-s + 1.31·29-s + 0.307·30-s − 0.156·31-s + 0.791·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8972464289\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8972464289\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 1.68T + 2T^{2} \) |
| 11 | \( 1 - 1.77T + 11T^{2} \) |
| 13 | \( 1 + 3.05T + 13T^{2} \) |
| 17 | \( 1 - 7.59T + 17T^{2} \) |
| 19 | \( 1 + 3.12T + 19T^{2} \) |
| 23 | \( 1 + 4.18T + 23T^{2} \) |
| 29 | \( 1 - 7.08T + 29T^{2} \) |
| 31 | \( 1 + 0.871T + 31T^{2} \) |
| 37 | \( 1 - 9.91T + 37T^{2} \) |
| 41 | \( 1 - 2.76T + 41T^{2} \) |
| 43 | \( 1 - 0.317T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 - 3.63T + 53T^{2} \) |
| 59 | \( 1 + 9.57T + 59T^{2} \) |
| 61 | \( 1 + 6.58T + 61T^{2} \) |
| 67 | \( 1 - 14.0T + 67T^{2} \) |
| 71 | \( 1 - 6.13T + 71T^{2} \) |
| 73 | \( 1 + 7.68T + 73T^{2} \) |
| 79 | \( 1 - 14.0T + 79T^{2} \) |
| 83 | \( 1 - 12.8T + 83T^{2} \) |
| 89 | \( 1 + 4.82T + 89T^{2} \) |
| 97 | \( 1 - 16.6T + 97T^{2} \) |
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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
−10.05893149754233455270106013744, −9.541367152812342220087987704393, −8.665750558679459997197831141393, −7.85722335892740383348586945139, −7.45521113594562803292315173128, −6.23719565817112738219833172257, −4.77371112936700725064848478109, −3.80780218327681429864951370159, −2.41970405860306252720019080212, −0.956888202047531280357347023545,
0.956888202047531280357347023545, 2.41970405860306252720019080212, 3.80780218327681429864951370159, 4.77371112936700725064848478109, 6.23719565817112738219833172257, 7.45521113594562803292315173128, 7.85722335892740383348586945139, 8.665750558679459997197831141393, 9.541367152812342220087987704393, 10.05893149754233455270106013744
