L(s) = 1 | + (0.974 + 0.222i)3-s + (0.781 + 0.623i)4-s + (−0.900 + 0.433i)7-s + (0.900 + 0.433i)9-s + (0.623 + 0.781i)12-s + (−0.974 + 0.222i)13-s + (0.222 + 0.974i)16-s + (1.33 − 1.33i)19-s + (−0.974 + 0.222i)21-s + (−0.433 + 0.900i)25-s + (0.781 + 0.623i)27-s + (−0.974 − 0.222i)28-s + (0.158 − 0.158i)31-s + (0.433 + 0.900i)36-s + (0.222 + 0.0250i)37-s + ⋯ |
L(s) = 1 | + (0.974 + 0.222i)3-s + (0.781 + 0.623i)4-s + (−0.900 + 0.433i)7-s + (0.900 + 0.433i)9-s + (0.623 + 0.781i)12-s + (−0.974 + 0.222i)13-s + (0.222 + 0.974i)16-s + (1.33 − 1.33i)19-s + (−0.974 + 0.222i)21-s + (−0.433 + 0.900i)25-s + (0.781 + 0.623i)27-s + (−0.974 − 0.222i)28-s + (0.158 − 0.158i)31-s + (0.433 + 0.900i)36-s + (0.222 + 0.0250i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.550 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.550 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.739381423\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.739381423\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.974 - 0.222i)T \) |
| 7 | \( 1 + (0.900 - 0.433i)T \) |
| 13 | \( 1 + (0.974 - 0.222i)T \) |
good | 2 | \( 1 + (-0.781 - 0.623i)T^{2} \) |
| 5 | \( 1 + (0.433 - 0.900i)T^{2} \) |
| 11 | \( 1 + (0.781 + 0.623i)T^{2} \) |
| 17 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 19 | \( 1 + (-1.33 + 1.33i)T - iT^{2} \) |
| 23 | \( 1 + (-0.222 - 0.974i)T^{2} \) |
| 29 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 31 | \( 1 + (-0.158 + 0.158i)T - iT^{2} \) |
| 37 | \( 1 + (-0.222 - 0.0250i)T + (0.974 + 0.222i)T^{2} \) |
| 41 | \( 1 + (0.433 - 0.900i)T^{2} \) |
| 43 | \( 1 + (0.846 - 0.193i)T + (0.900 - 0.433i)T^{2} \) |
| 47 | \( 1 + (-0.781 - 0.623i)T^{2} \) |
| 53 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 59 | \( 1 + (0.433 + 0.900i)T^{2} \) |
| 61 | \( 1 + (-0.974 + 0.777i)T + (0.222 - 0.974i)T^{2} \) |
| 67 | \( 1 + (0.752 + 0.752i)T + iT^{2} \) |
| 71 | \( 1 + (-0.974 + 0.222i)T^{2} \) |
| 73 | \( 1 + (0.656 + 1.87i)T + (-0.781 + 0.623i)T^{2} \) |
| 79 | \( 1 + 1.56T + T^{2} \) |
| 83 | \( 1 + (-0.781 + 0.623i)T^{2} \) |
| 89 | \( 1 + (0.781 - 0.623i)T^{2} \) |
| 97 | \( 1 + (0.467 - 0.467i)T - iT^{2} \) |
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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
−9.477538739061329414911210771243, −8.827152706876263208370376711743, −7.86581476180545101254336200292, −7.22031320219124357386606684437, −6.70206032561965577231593927628, −5.49703324318897627077363298724, −4.45533101256692231493689267643, −3.27272558117540331697914247797, −2.91020260898510177838283776701, −1.91437546004908313680180866383,
1.23529184364967409014333212616, 2.43757927206723365511463632435, 3.16413529970942665065303880208, 4.12519652132649365308638608194, 5.39195373758009074612437635440, 6.23866723766027513199435601230, 7.10199877149871771849246859473, 7.52175640325950757850126017278, 8.412210149972776789830665235243, 9.534821452596728666802333116675