| 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
| \(L(1)\) |
\(\approx\) |
\(0.6971471080\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6971471080\) |
| \(L(\frac{3}{2})\) |
|
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
−10.15681149205617680676131415916, −9.604443606007077475518533216079, −8.721266304557922228102935440418, −8.077006554265791860081936382042, −6.80901959666663896447911738375, −6.31689482484344812180830206007, −4.98714025267253925096677657205, −4.01369742879643642444167061227, −2.16043990288743640519348600018, −0.887070391376747885592512794311,
0.887070391376747885592512794311, 2.16043990288743640519348600018, 4.01369742879643642444167061227, 4.98714025267253925096677657205, 6.31689482484344812180830206007, 6.80901959666663896447911738375, 8.077006554265791860081936382042, 8.721266304557922228102935440418, 9.604443606007077475518533216079, 10.15681149205617680676131415916
