L(s) = 1 | + (0.5 − 0.866i)4-s + (0.5 − 0.866i)7-s + 1.73i·13-s + (−0.499 − 0.866i)16-s + (1.5 − 0.866i)19-s + (−0.499 − 0.866i)28-s + (−0.5 − 0.866i)37-s − 2·43-s + (−0.499 − 0.866i)49-s + (1.49 + 0.866i)52-s + (−1.5 + 0.866i)61-s − 0.999·64-s + (0.5 − 0.866i)67-s + (1.5 + 0.866i)73-s − 1.73i·76-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)4-s + (0.5 − 0.866i)7-s + 1.73i·13-s + (−0.499 − 0.866i)16-s + (1.5 − 0.866i)19-s + (−0.499 − 0.866i)28-s + (−0.5 − 0.866i)37-s − 2·43-s + (−0.499 − 0.866i)49-s + (1.49 + 0.866i)52-s + (−1.5 + 0.866i)61-s − 0.999·64-s + (0.5 − 0.866i)67-s + (1.5 + 0.866i)73-s − 1.73i·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.322807237\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.322807237\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
good | 2 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 - 1.73iT - T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 2T + T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 - 1.73iT - T^{2} \) |
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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
−9.565360558033900248364380465734, −8.909757361405363848103438008072, −7.68824493082618607930019820666, −6.99329281199656667288320251889, −6.47562218671681984953798436746, −5.25775626116668737001177135895, −4.66840421544948942162143250317, −3.56203413978644367886573327063, −2.16877986192885348746052775847, −1.20230275749537431270299909491,
1.68604251927303734649373186138, 2.96215741090627392784972702705, 3.43807187385475128884232854567, 4.91471620087078538433950296498, 5.60313709408012586111736879051, 6.51519360316686909037643447562, 7.65024385460216583404467428725, 8.010216691182826770194000175682, 8.715781220519617269067992459067, 9.759633078010087114845882513777