L(s) = 1 | + (1.30 − 0.951i)2-s + (0.309 + 0.951i)3-s + (0.500 − 1.53i)4-s + (−0.5 − 0.363i)5-s + (1.30 + 0.951i)6-s + (−0.309 − 0.951i)8-s + (−0.809 + 0.587i)9-s − 10-s + (−0.809 − 0.587i)11-s + 1.61·12-s + (−0.809 + 0.587i)13-s + (0.190 − 0.587i)15-s + (−0.499 + 1.53i)18-s + (−0.809 + 0.587i)20-s − 1.61·22-s + ⋯ |
L(s) = 1 | + (1.30 − 0.951i)2-s + (0.309 + 0.951i)3-s + (0.500 − 1.53i)4-s + (−0.5 − 0.363i)5-s + (1.30 + 0.951i)6-s + (−0.309 − 0.951i)8-s + (−0.809 + 0.587i)9-s − 10-s + (−0.809 − 0.587i)11-s + 1.61·12-s + (−0.809 + 0.587i)13-s + (0.190 − 0.587i)15-s + (−0.499 + 1.53i)18-s + (−0.809 + 0.587i)20-s − 1.61·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 + 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 + 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.526638123\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.526638123\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.309 - 0.951i)T \) |
| 11 | \( 1 + (0.809 + 0.587i)T \) |
| 13 | \( 1 + (0.809 - 0.587i)T \) |
good | 2 | \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 5 | \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 7 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 19 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 31 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 41 | \( 1 + (-0.618 - 1.90i)T + (-0.809 + 0.587i)T^{2} \) |
| 43 | \( 1 - 0.618T + T^{2} \) |
| 47 | \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 53 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 59 | \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (1.61 + 1.17i)T + (0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 79 | \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 89 | \( 1 - 2T + T^{2} \) |
| 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
−11.49664824254612152427935132512, −10.56831580948536694838757312909, −9.889951305976609608343033165024, −8.716194392510235710063036383830, −7.75808725216106026852526261124, −6.03462950567384111738172190494, −4.98623120538750706106171505842, −4.40620994771257051691464826572, −3.36516045675703410611326529525, −2.38355047805118523514237716335,
2.53536426066171617188298728752, 3.58722485375291194822162730118, 4.91429579640511628840861985634, 5.81474420226351671011310507734, 6.88297696684549833608361133753, 7.54582421722621227070330404234, 8.040127292498926733022239963097, 9.535548659844534397633048761481, 10.88548153941724394379853471341, 11.97963818612987967760176749008