L(s) = 1 | + (−0.433 − 0.900i)2-s + (−0.623 + 0.781i)4-s + (−1.62 + 0.781i)5-s + (0.974 + 0.222i)8-s + (0.222 − 0.974i)9-s + (1.40 + 1.12i)10-s + (−0.0990 − 0.433i)13-s + (−0.222 − 0.974i)16-s + 1.24i·17-s + (−0.974 + 0.222i)18-s + (0.400 − 1.75i)20-s + (1.40 − 1.75i)25-s + (−0.347 + 0.277i)26-s + (−0.781 + 0.623i)32-s + (1.12 − 0.541i)34-s + ⋯ |
L(s) = 1 | + (−0.433 − 0.900i)2-s + (−0.623 + 0.781i)4-s + (−1.62 + 0.781i)5-s + (0.974 + 0.222i)8-s + (0.222 − 0.974i)9-s + (1.40 + 1.12i)10-s + (−0.0990 − 0.433i)13-s + (−0.222 − 0.974i)16-s + 1.24i·17-s + (−0.974 + 0.222i)18-s + (0.400 − 1.75i)20-s + (1.40 − 1.75i)25-s + (−0.347 + 0.277i)26-s + (−0.781 + 0.623i)32-s + (1.12 − 0.541i)34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.354 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.354 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3640446779\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3640446779\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.433 + 0.900i)T \) |
| 29 | \( 1 \) |
good | 3 | \( 1 + (-0.222 + 0.974i)T^{2} \) |
| 5 | \( 1 + (1.62 - 0.781i)T + (0.623 - 0.781i)T^{2} \) |
| 7 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 11 | \( 1 + (-0.900 + 0.433i)T^{2} \) |
| 13 | \( 1 + (0.0990 + 0.433i)T + (-0.900 + 0.433i)T^{2} \) |
| 17 | \( 1 - 1.24iT - T^{2} \) |
| 19 | \( 1 + (-0.222 - 0.974i)T^{2} \) |
| 23 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 31 | \( 1 + (0.623 - 0.781i)T^{2} \) |
| 37 | \( 1 + (0.433 + 0.0990i)T + (0.900 + 0.433i)T^{2} \) |
| 41 | \( 1 - 0.445iT - T^{2} \) |
| 43 | \( 1 + (0.623 + 0.781i)T^{2} \) |
| 47 | \( 1 + (-0.900 + 0.433i)T^{2} \) |
| 53 | \( 1 + (1.12 - 0.541i)T + (0.623 - 0.781i)T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + (0.974 - 0.777i)T + (0.222 - 0.974i)T^{2} \) |
| 67 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 71 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 73 | \( 1 + (0.541 - 1.12i)T + (-0.623 - 0.781i)T^{2} \) |
| 79 | \( 1 + (-0.900 - 0.433i)T^{2} \) |
| 83 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 89 | \( 1 + (-0.781 - 1.62i)T + (-0.623 + 0.781i)T^{2} \) |
| 97 | \( 1 + (-0.347 - 0.277i)T + (0.222 + 0.974i)T^{2} \) |
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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
−8.916546625044725605778307876403, −8.182259765892579807854263157245, −7.68719438214083422834947219291, −6.94968217154760264765485380396, −6.10991517259585369663690088085, −4.69720191953544939122321143846, −3.92947927598592384093239036030, −3.46519420895721447716586032361, −2.67683019283055385883663717870, −1.18293805394837599033848754942,
0.29456939687862371732963573272, 1.70844425349360091810439349902, 3.30968407040227482464851093041, 4.42388033450276833221715395189, 4.76269906501147594444998560166, 5.49577663605591502281712530634, 6.74699329801896956769363340906, 7.36892087501327380884214829338, 7.84696307306432616092557942174, 8.488538517601639198870950552108