L(s) = 1 | + (0.680 + 0.733i)2-s + (−0.0747 + 0.997i)4-s + (−0.215 + 0.548i)5-s + (0.365 + 0.930i)7-s + (−0.781 + 0.623i)8-s + (−0.548 + 0.215i)10-s + (0.0440 − 0.142i)11-s + (−0.433 + 0.900i)14-s + (−0.988 − 0.149i)16-s + (−0.531 − 0.255i)20-s + (0.134 − 0.0648i)22-s + (0.478 + 0.443i)25-s + (−0.955 + 0.294i)28-s + (−0.636 + 1.32i)29-s + (0.258 + 0.149i)31-s + (−0.563 − 0.826i)32-s + ⋯ |
L(s) = 1 | + (0.680 + 0.733i)2-s + (−0.0747 + 0.997i)4-s + (−0.215 + 0.548i)5-s + (0.365 + 0.930i)7-s + (−0.781 + 0.623i)8-s + (−0.548 + 0.215i)10-s + (0.0440 − 0.142i)11-s + (−0.433 + 0.900i)14-s + (−0.988 − 0.149i)16-s + (−0.531 − 0.255i)20-s + (0.134 − 0.0648i)22-s + (0.478 + 0.443i)25-s + (−0.955 + 0.294i)28-s + (−0.636 + 1.32i)29-s + (0.258 + 0.149i)31-s + (−0.563 − 0.826i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.918 - 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.918 - 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.615137026\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.615137026\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.680 - 0.733i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.365 - 0.930i)T \) |
good | 5 | \( 1 + (0.215 - 0.548i)T + (-0.733 - 0.680i)T^{2} \) |
| 11 | \( 1 + (-0.0440 + 0.142i)T + (-0.826 - 0.563i)T^{2} \) |
| 13 | \( 1 + (-0.900 + 0.433i)T^{2} \) |
| 17 | \( 1 + (-0.365 + 0.930i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.365 + 0.930i)T^{2} \) |
| 29 | \( 1 + (0.636 - 1.32i)T + (-0.623 - 0.781i)T^{2} \) |
| 31 | \( 1 + (-0.258 - 0.149i)T + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.988 - 0.149i)T^{2} \) |
| 41 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 43 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 47 | \( 1 + (-0.0747 + 0.997i)T^{2} \) |
| 53 | \( 1 + (0.997 + 0.0747i)T + (0.988 + 0.149i)T^{2} \) |
| 59 | \( 1 + (0.632 + 1.61i)T + (-0.733 + 0.680i)T^{2} \) |
| 61 | \( 1 + (-0.988 + 0.149i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (0.623 - 0.781i)T^{2} \) |
| 73 | \( 1 + (-1.06 + 1.14i)T + (-0.0747 - 0.997i)T^{2} \) |
| 79 | \( 1 + (-0.365 - 0.632i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.302 + 1.32i)T + (-0.900 - 0.433i)T^{2} \) |
| 89 | \( 1 + (-0.826 + 0.563i)T^{2} \) |
| 97 | \( 1 - 1.12iT - T^{2} \) |
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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
−8.923308404484535798663182995072, −8.157460034593366602971748161527, −7.53726402580705340082349004507, −6.72992490510583189480847324541, −6.13650107669862560520320966938, −5.25294986748780107871585137514, −4.76060185607555365696141071082, −3.56218407339029626751685332197, −3.00937446419487145886004858330, −1.92540505257367959928312350022,
0.75454546426788277584859318521, 1.79888648506529794876970136757, 2.88613553327798323000481021694, 3.95883392524899895423340287946, 4.40412964504198269077469906108, 5.14882524126645114388934012281, 6.03471124986440337743003287001, 6.84395346110445616804954621091, 7.70757001660841399091388852471, 8.458044094835306988895239928094