L(s) = 1 | + (−0.309 + 0.951i)2-s + (−0.951 − 0.690i)3-s + (−0.809 − 0.587i)4-s + (−0.951 − 0.309i)5-s + (0.951 − 0.690i)6-s − 7-s + (0.809 − 0.587i)8-s + (0.118 + 0.363i)9-s + (0.587 − 0.809i)10-s + (0.363 + 1.11i)12-s + (−0.363 − 1.11i)13-s + (0.309 − 0.951i)14-s + (0.690 + 0.951i)15-s + (0.309 + 0.951i)16-s − 0.381·18-s + ⋯ |
L(s) = 1 | + (−0.309 + 0.951i)2-s + (−0.951 − 0.690i)3-s + (−0.809 − 0.587i)4-s + (−0.951 − 0.309i)5-s + (0.951 − 0.690i)6-s − 7-s + (0.809 − 0.587i)8-s + (0.118 + 0.363i)9-s + (0.587 − 0.809i)10-s + (0.363 + 1.11i)12-s + (−0.363 − 1.11i)13-s + (0.309 − 0.951i)14-s + (0.690 + 0.951i)15-s + (0.309 + 0.951i)16-s − 0.381·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1641909623\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1641909623\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.309 - 0.951i)T \) |
| 5 | \( 1 + (0.951 + 0.309i)T \) |
| 7 | \( 1 + T \) |
good | 3 | \( 1 + (0.951 + 0.690i)T + (0.309 + 0.951i)T^{2} \) |
| 11 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 13 | \( 1 + (0.363 + 1.11i)T + (-0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 19 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 29 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 41 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 53 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (-0.587 - 1.80i)T + (-0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (0.587 - 1.80i)T + (-0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 71 | \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 79 | \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (1.53 - 1.11i)T + (0.309 - 0.951i)T^{2} \) |
| 89 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 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
−9.929853087427870369365497987937, −9.092945875031152543799271022875, −8.195520642281225773788377613018, −7.32469654630836265254292226433, −7.01796097134058085074293548121, −5.81426637821549381492400459473, −5.56431089845192899937052131260, −4.29908048168071183420264808618, −3.25653215320504421441930454776, −1.04958346167430216007517007516,
0.21081003325642543076509382466, 2.35155292541466349688018106358, 3.48607617149681610483187808566, 4.28461936298207826056382780135, 4.89403118273409426741200302999, 6.21809941591680084889248442738, 7.00804181679847158553252432590, 8.071817211591496301052886011327, 8.920161625485446722296699071497, 9.818296457745604142372421733040