L(s) = 1 | − 0.732i·3-s + i·5-s + 1.26·7-s + 2.46·9-s − 3.46i·11-s + 3.46i·13-s + 0.732·15-s + 3.46·17-s − 2i·19-s − 0.928i·21-s + 8.19·23-s − 25-s − 4i·27-s − 9.46·31-s − 2.53·33-s + ⋯ |
L(s) = 1 | − 0.422i·3-s + 0.447i·5-s + 0.479·7-s + 0.821·9-s − 1.04i·11-s + 0.960i·13-s + 0.189·15-s + 0.840·17-s − 0.458i·19-s − 0.202i·21-s + 1.70·23-s − 0.200·25-s − 0.769i·27-s − 1.69·31-s − 0.441·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.45274 - 0.191257i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.45274 - 0.191257i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - iT \) |
good | 3 | \( 1 + 0.732iT - 3T^{2} \) |
| 7 | \( 1 - 1.26T + 7T^{2} \) |
| 11 | \( 1 + 3.46iT - 11T^{2} \) |
| 13 | \( 1 - 3.46iT - 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 + 2iT - 19T^{2} \) |
| 23 | \( 1 - 8.19T + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 9.46T + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 - 2.53T + 41T^{2} \) |
| 43 | \( 1 - 10.1iT - 43T^{2} \) |
| 47 | \( 1 + 8.19T + 47T^{2} \) |
| 53 | \( 1 - 10.3iT - 53T^{2} \) |
| 59 | \( 1 - 6iT - 59T^{2} \) |
| 61 | \( 1 + 12.9iT - 61T^{2} \) |
| 67 | \( 1 - 10.1iT - 67T^{2} \) |
| 71 | \( 1 + 4.39T + 71T^{2} \) |
| 73 | \( 1 + 14.3T + 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 - 4.73iT - 83T^{2} \) |
| 89 | \( 1 - 0.928T + 89T^{2} \) |
| 97 | \( 1 + 6.39T + 97T^{2} \) |
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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
−11.36192184800916509733603003567, −10.97084909562719418704448052929, −9.683300554744843908240160008195, −8.779527784589439743315591992253, −7.58721003950842959170604791400, −6.91245819544501300939434490426, −5.76482792120193023122931491328, −4.46653452313539574964867108407, −3.10347785620104204351078802474, −1.44953091408692283366044663361,
1.54913821355366333262453587199, 3.44363050086677859955276502146, 4.71955098095264012975912458241, 5.40921528857901648583958074354, 7.03653098665293352722038615522, 7.81485946296382063015285684623, 8.965714159453974983371317260698, 9.912169106859513119862037278186, 10.55200951368201821154770583021, 11.69029099573638836462897871724