L(s) = 1 | + (−0.587 − 0.190i)2-s + (0.809 − 0.587i)3-s + (−0.5 − 0.363i)4-s + (−0.587 + 0.809i)5-s + (−0.587 + 0.190i)6-s + (0.309 + 0.951i)7-s + (0.587 + 0.809i)8-s + (0.309 − 0.951i)9-s + (0.5 − 0.363i)10-s + (0.951 − 1.30i)11-s − 0.618·12-s + (−0.309 − 0.951i)13-s − 0.618i·14-s + i·15-s + (1.53 − 0.5i)17-s + (−0.363 + 0.5i)18-s + ⋯ |
L(s) = 1 | + (−0.587 − 0.190i)2-s + (0.809 − 0.587i)3-s + (−0.5 − 0.363i)4-s + (−0.587 + 0.809i)5-s + (−0.587 + 0.190i)6-s + (0.309 + 0.951i)7-s + (0.587 + 0.809i)8-s + (0.309 − 0.951i)9-s + (0.5 − 0.363i)10-s + (0.951 − 1.30i)11-s − 0.618·12-s + (−0.309 − 0.951i)13-s − 0.618i·14-s + i·15-s + (1.53 − 0.5i)17-s + (−0.363 + 0.5i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 633 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.751 + 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 633 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.751 + 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7896170618\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7896170618\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.809 + 0.587i)T \) |
| 211 | \( 1 + (-0.309 - 0.951i)T \) |
good | 2 | \( 1 + (0.587 + 0.190i)T + (0.809 + 0.587i)T^{2} \) |
| 5 | \( 1 + (0.587 - 0.809i)T + (-0.309 - 0.951i)T^{2} \) |
| 7 | \( 1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2} \) |
| 11 | \( 1 + (-0.951 + 1.30i)T + (-0.309 - 0.951i)T^{2} \) |
| 13 | \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (-1.53 + 0.5i)T + (0.809 - 0.587i)T^{2} \) |
| 19 | \( 1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 + (0.951 + 0.309i)T + (0.809 + 0.587i)T^{2} \) |
| 29 | \( 1 + (0.587 - 0.190i)T + (0.809 - 0.587i)T^{2} \) |
| 31 | \( 1 - 0.618T + T^{2} \) |
| 37 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 41 | \( 1 + (0.951 - 1.30i)T + (-0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( 1 + (-0.587 + 0.190i)T + (0.809 - 0.587i)T^{2} \) |
| 53 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 59 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + (-0.363 - 0.5i)T + (-0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.363 + 0.5i)T + (-0.309 - 0.951i)T^{2} \) |
| 89 | \( 1 + (-0.951 + 0.309i)T + (0.809 - 0.587i)T^{2} \) |
| 97 | \( 1 + (0.309 - 0.951i)T^{2} \) |
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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
−10.48327844638390701405541912317, −9.749293794376488307719818789909, −8.840198935546719210961999846036, −8.122740057451222617644694114300, −7.64432463450359418652160279602, −6.23054457102105323372141301629, −5.41904430229142979059166833678, −3.68982596198123419253566341173, −2.90008629134652037831146540343, −1.33545731319641007077210955580,
1.55753865669831610511655494103, 3.68582118115226648217656685882, 4.25708827139220254382524781558, 4.89769086620307088203601061944, 7.00711151655427867585610853676, 7.63080925363054826637619306619, 8.307436141523274116357340759762, 9.236120068661618813362696383699, 9.707256983215058832184288000992, 10.47876716676265304770496524190