L(s) = 1 | + 1.41·2-s + 1.00·4-s + 5-s + 1.41·10-s − 1.41·13-s − 0.999·16-s − 19-s + 1.00·20-s + 25-s − 2.00·26-s − 1.41·32-s + 1.41·37-s − 1.41·38-s + 49-s + 1.41·50-s − 1.41·52-s − 1.41·53-s − 1.00·64-s − 1.41·65-s − 1.41·67-s + 2.00·74-s − 1.00·76-s − 0.999·80-s − 95-s + 1.41·97-s + 1.41·98-s + 1.00·100-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 1.00·4-s + 5-s + 1.41·10-s − 1.41·13-s − 0.999·16-s − 19-s + 1.00·20-s + 25-s − 2.00·26-s − 1.41·32-s + 1.41·37-s − 1.41·38-s + 49-s + 1.41·50-s − 1.41·52-s − 1.41·53-s − 1.00·64-s − 1.41·65-s − 1.41·67-s + 2.00·74-s − 1.00·76-s − 0.999·80-s − 95-s + 1.41·97-s + 1.41·98-s + 1.00·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.064185105\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.064185105\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 + T \) |
good | 2 | \( 1 - 1.41T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + 1.41T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - 1.41T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + 1.41T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + 1.41T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - 1.41T + T^{2} \) |
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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
−10.45792011474811071120776346473, −9.623923462398388958674827630161, −8.855992603957737578040280210684, −7.53874129702747483862315240467, −6.54735645376233508662178637077, −5.88672294809600988220141059621, −4.98490716495764179214499644337, −4.31432945832676453282242500406, −2.94017852514384129976399413879, −2.13367658955366694371033754489,
2.13367658955366694371033754489, 2.94017852514384129976399413879, 4.31432945832676453282242500406, 4.98490716495764179214499644337, 5.88672294809600988220141059621, 6.54735645376233508662178637077, 7.53874129702747483862315240467, 8.855992603957737578040280210684, 9.623923462398388958674827630161, 10.45792011474811071120776346473