L(s) = 1 | + 1.41·2-s − 3-s + 1.00·4-s + 5-s − 1.41·6-s + 9-s + 1.41·10-s − 1.00·12-s − 15-s − 0.999·16-s + 1.41·18-s + 1.41·19-s + 1.00·20-s − 1.41·23-s + 25-s − 27-s − 1.41·30-s − 1.41·31-s − 1.41·32-s + 1.00·36-s + 2.00·38-s + 45-s − 2.00·46-s + 0.999·48-s + 1.41·50-s − 1.41·53-s + ⋯ |
L(s) = 1 | + 1.41·2-s − 3-s + 1.00·4-s + 5-s − 1.41·6-s + 9-s + 1.41·10-s − 1.00·12-s − 15-s − 0.999·16-s + 1.41·18-s + 1.41·19-s + 1.00·20-s − 1.41·23-s + 25-s − 27-s − 1.41·30-s − 1.41·31-s − 1.41·32-s + 1.00·36-s + 2.00·38-s + 45-s − 2.00·46-s + 0.999·48-s + 1.41·50-s − 1.41·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.607748460\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.607748460\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 1.41T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - 1.41T + T^{2} \) |
| 23 | \( 1 + 1.41T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + 1.41T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + 1.41T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + 1.41T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−10.84450122207702245493234375661, −9.890099789933889773189314341736, −9.190294197280203749628783938272, −7.58400158631477052659106977324, −6.59345587307017599841416552965, −5.83921999502597708888875756938, −5.33204746641583071004976068193, −4.43411112815052962302483509080, −3.28390921012292782619901431060, −1.82901295326634746793103000389,
1.82901295326634746793103000389, 3.28390921012292782619901431060, 4.43411112815052962302483509080, 5.33204746641583071004976068193, 5.83921999502597708888875756938, 6.59345587307017599841416552965, 7.58400158631477052659106977324, 9.190294197280203749628783938272, 9.890099789933889773189314341736, 10.84450122207702245493234375661