L(s) = 1 | + (−0.406 − 0.913i)2-s + (−0.309 − 0.951i)3-s + (−0.669 + 0.743i)4-s + (−0.743 − 0.669i)5-s + (−0.743 + 0.669i)6-s + (0.866 + 0.5i)7-s + (0.951 + 0.309i)8-s + (−0.809 + 0.587i)9-s + (−0.309 + 0.951i)10-s + (1.47 − 0.658i)11-s + (0.913 + 0.406i)12-s + (−0.251 + 0.564i)13-s + (0.104 − 0.994i)14-s + (−0.406 + 0.913i)15-s + (−0.104 − 0.994i)16-s + (0.309 − 0.951i)17-s + ⋯ |
L(s) = 1 | + (−0.406 − 0.913i)2-s + (−0.309 − 0.951i)3-s + (−0.669 + 0.743i)4-s + (−0.743 − 0.669i)5-s + (−0.743 + 0.669i)6-s + (0.866 + 0.5i)7-s + (0.951 + 0.309i)8-s + (−0.809 + 0.587i)9-s + (−0.309 + 0.951i)10-s + (1.47 − 0.658i)11-s + (0.913 + 0.406i)12-s + (−0.251 + 0.564i)13-s + (0.104 − 0.994i)14-s + (−0.406 + 0.913i)15-s + (−0.104 − 0.994i)16-s + (0.309 − 0.951i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.946 + 0.322i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.946 + 0.322i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7330618682\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7330618682\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.406 + 0.913i)T \) |
| 3 | \( 1 + (0.309 + 0.951i)T \) |
| 5 | \( 1 + (0.743 + 0.669i)T \) |
good | 7 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-1.47 + 0.658i)T + (0.669 - 0.743i)T^{2} \) |
| 13 | \( 1 + (0.251 - 0.564i)T + (-0.669 - 0.743i)T^{2} \) |
| 17 | \( 1 + (-0.309 + 0.951i)T + (-0.809 - 0.587i)T^{2} \) |
| 19 | \( 1 + (-0.309 + 0.951i)T + (-0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 + (0.994 + 0.104i)T + (0.978 + 0.207i)T^{2} \) |
| 29 | \( 1 + (-0.336 + 1.58i)T + (-0.913 - 0.406i)T^{2} \) |
| 31 | \( 1 + (0.207 + 0.978i)T + (-0.913 + 0.406i)T^{2} \) |
| 37 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.564 - 0.251i)T + (0.669 + 0.743i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.336 - 1.58i)T + (-0.913 - 0.406i)T^{2} \) |
| 53 | \( 1 + (-0.587 + 0.190i)T + (0.809 - 0.587i)T^{2} \) |
| 59 | \( 1 + (0.669 + 0.743i)T^{2} \) |
| 61 | \( 1 + (0.406 + 0.913i)T + (-0.669 + 0.743i)T^{2} \) |
| 67 | \( 1 + (-0.604 + 0.128i)T + (0.913 - 0.406i)T^{2} \) |
| 71 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 73 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 79 | \( 1 + (0.336 - 1.58i)T + (-0.913 - 0.406i)T^{2} \) |
| 83 | \( 1 + (1.08 + 1.20i)T + (-0.104 + 0.994i)T^{2} \) |
| 89 | \( 1 + (0.809 + 0.587i)T + (0.309 + 0.951i)T^{2} \) |
| 97 | \( 1 + (0.913 + 0.406i)T^{2} \) |
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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
−9.140485538142870165361680412242, −8.252218670766306445235974241805, −7.889538345398627015198402540930, −6.95371361050697985613941741191, −5.86330553828761793216228524576, −4.78735267386306405098608064330, −4.13869564844301023449704657488, −2.83594121272150520737359005633, −1.75363974006920692437642677306, −0.75682764442575381095211039185,
1.46960511835382234804471289710, 3.62239472165002986123489618705, 4.05520679928668020146280992377, 4.96741702682547005173777477211, 5.84803455898376105404058816986, 6.75880450537414839957246206402, 7.41170878709325488905033896451, 8.304663124967655419270751934707, 8.826023115894607656482587320499, 9.996706862335453383891673514770