L(s) = 1 | + 1.10·2-s − 0.783·4-s + (−0.826 − 2.07i)5-s + (−1.13 + 2.38i)7-s − 3.07·8-s + (−0.911 − 2.29i)10-s + 2.51i·11-s + 6.14·13-s + (−1.25 + 2.63i)14-s − 1.81·16-s + 2.69i·17-s + 2.85i·19-s + (0.647 + 1.62i)20-s + 2.77i·22-s − 4.90·23-s + ⋯ |
L(s) = 1 | + 0.779·2-s − 0.391·4-s + (−0.369 − 0.929i)5-s + (−0.429 + 0.902i)7-s − 1.08·8-s + (−0.288 − 0.724i)10-s + 0.758i·11-s + 1.70·13-s + (−0.335 + 0.704i)14-s − 0.454·16-s + 0.653i·17-s + 0.655i·19-s + (0.144 + 0.363i)20-s + 0.591i·22-s − 1.02·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0657 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0657 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.944286 + 0.884127i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.944286 + 0.884127i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (0.826 + 2.07i)T \) |
| 7 | \( 1 + (1.13 - 2.38i)T \) |
good | 2 | \( 1 - 1.10T + 2T^{2} \) |
| 11 | \( 1 - 2.51iT - 11T^{2} \) |
| 13 | \( 1 - 6.14T + 13T^{2} \) |
| 17 | \( 1 - 2.69iT - 17T^{2} \) |
| 19 | \( 1 - 2.85iT - 19T^{2} \) |
| 23 | \( 1 + 4.90T + 23T^{2} \) |
| 29 | \( 1 - 3.39iT - 29T^{2} \) |
| 31 | \( 1 - 3.91iT - 31T^{2} \) |
| 37 | \( 1 - 6.43iT - 37T^{2} \) |
| 41 | \( 1 + 0.647T + 41T^{2} \) |
| 43 | \( 1 - 10.8iT - 43T^{2} \) |
| 47 | \( 1 + 0.270iT - 47T^{2} \) |
| 53 | \( 1 - 12.8T + 53T^{2} \) |
| 59 | \( 1 + 14.4T + 59T^{2} \) |
| 61 | \( 1 + 8.83iT - 61T^{2} \) |
| 67 | \( 1 - 6.34iT - 67T^{2} \) |
| 71 | \( 1 + 13.3iT - 71T^{2} \) |
| 73 | \( 1 - 13.0T + 73T^{2} \) |
| 79 | \( 1 + 8.30T + 79T^{2} \) |
| 83 | \( 1 + 8.24iT - 83T^{2} \) |
| 89 | \( 1 + 3.46T + 89T^{2} \) |
| 97 | \( 1 + 13.2T + 97T^{2} \) |
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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
−10.11021412315821397357485723244, −9.265546189531542227791681955665, −8.578550670906035159897301096862, −8.037946319065933613514607712792, −6.40253476521365167149798241337, −5.83634110679531740439911839985, −4.92641507467209480734798444764, −4.04105976307732400475673425251, −3.26643274199662165864685111070, −1.55979015064664014658377457301,
0.49535630661058402374216553486, 2.75089958232053548563241448282, 3.81853947347056916423086093174, 4.03548722847328265776830253958, 5.61539513183307433825480721309, 6.26046016023606322378173510988, 7.12205529957407694863673570927, 8.146897816435893433360120569914, 8.992286685312827441748989342729, 9.965388319109365529476706877227