| L(s) = 1 | − 1.73i·2-s + (−1.41 + i)3-s − 0.999·4-s + (1.73 + 2.44i)6-s + (−2.44 + i)7-s − 1.73i·8-s + (1.00 − 2.82i)9-s + 2.82i·11-s + (1.41 − 0.999i)12-s + 4i·13-s + (1.73 + 4.24i)14-s − 5·16-s − 2.82·17-s + (−4.89 − 1.73i)18-s + (2.46 − 3.86i)21-s + 4.89·22-s + ⋯ |
| L(s) = 1 | − 1.22i·2-s + (−0.816 + 0.577i)3-s − 0.499·4-s + (0.707 + 0.999i)6-s + (−0.925 + 0.377i)7-s − 0.612i·8-s + (0.333 − 0.942i)9-s + 0.852i·11-s + (0.408 − 0.288i)12-s + 1.10i·13-s + (0.462 + 1.13i)14-s − 1.25·16-s − 0.685·17-s + (−1.15 − 0.408i)18-s + (0.537 − 0.843i)21-s + 1.04·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.537 - 0.843i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.537 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.507870 + 0.278466i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.507870 + 0.278466i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.41 - i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.44 - i)T \) |
| good | 2 | \( 1 + 1.73iT - 2T^{2} \) |
| 11 | \( 1 - 2.82iT - 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 + 2.82T + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 3.46iT - 23T^{2} \) |
| 29 | \( 1 - 5.65iT - 29T^{2} \) |
| 31 | \( 1 - 9.79iT - 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 3.46T + 41T^{2} \) |
| 43 | \( 1 - 4.89T + 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 6.92T + 59T^{2} \) |
| 61 | \( 1 + 9.79iT - 61T^{2} \) |
| 67 | \( 1 - 4.89T + 67T^{2} \) |
| 71 | \( 1 - 2.82iT - 71T^{2} \) |
| 73 | \( 1 + 8iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 2.82T + 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 - 8iT - 97T^{2} \) |
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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
−10.98810205377430434646893258859, −10.29402773762022808284365950999, −9.466144426524790327033491114834, −9.027436397531552309515880943751, −6.97353604084205840388794881896, −6.52640397618182810756026004890, −5.10368955958346986016168066349, −4.10966826692915415762069196150, −3.13270138117997919483161469832, −1.67990065003813882294903981707,
0.36422796138032839689532786261, 2.62678131289774641261413808408, 4.34921749087422178565706792364, 5.69112696278404865574038099762, 6.08982515924491753361238961010, 6.94254914315368758352926512651, 7.76609646723038776318040430757, 8.527026226510896588755644854835, 9.842119214266647048228940807495, 10.82537480129378212773993164572
