L(s) = 1 | + (−0.766 + 0.642i)5-s + (0.939 + 0.342i)9-s + (−0.592 − 1.62i)13-s + (0.439 − 1.20i)17-s + (0.173 − 0.984i)25-s + (0.939 + 1.62i)29-s + (0.939 − 0.342i)37-s + (1.43 − 0.524i)41-s + (−0.939 + 0.342i)45-s + (−0.173 − 0.984i)49-s + (1.11 + 1.32i)53-s + (−1.76 + 0.642i)61-s + (1.5 + 0.866i)65-s + 1.73i·73-s + (0.766 + 0.642i)81-s + ⋯ |
L(s) = 1 | + (−0.766 + 0.642i)5-s + (0.939 + 0.342i)9-s + (−0.592 − 1.62i)13-s + (0.439 − 1.20i)17-s + (0.173 − 0.984i)25-s + (0.939 + 1.62i)29-s + (0.939 − 0.342i)37-s + (1.43 − 0.524i)41-s + (−0.939 + 0.342i)45-s + (−0.173 − 0.984i)49-s + (1.11 + 1.32i)53-s + (−1.76 + 0.642i)61-s + (1.5 + 0.866i)65-s + 1.73i·73-s + (0.766 + 0.642i)81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.138i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.138i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.141453372\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.141453372\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.766 - 0.642i)T \) |
| 37 | \( 1 + (-0.939 + 0.342i)T \) |
good | 3 | \( 1 + (-0.939 - 0.342i)T^{2} \) |
| 7 | \( 1 + (0.173 + 0.984i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.592 + 1.62i)T + (-0.766 + 0.642i)T^{2} \) |
| 17 | \( 1 + (-0.439 + 1.20i)T + (-0.766 - 0.642i)T^{2} \) |
| 19 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 41 | \( 1 + (-1.43 + 0.524i)T + (0.766 - 0.642i)T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-1.11 - 1.32i)T + (-0.173 + 0.984i)T^{2} \) |
| 59 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 61 | \( 1 + (1.76 - 0.642i)T + (0.766 - 0.642i)T^{2} \) |
| 67 | \( 1 + (0.173 + 0.984i)T^{2} \) |
| 71 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 73 | \( 1 - 1.73iT - T^{2} \) |
| 79 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 83 | \( 1 + (0.766 + 0.642i)T^{2} \) |
| 89 | \( 1 + (-1.17 + 0.984i)T + (0.173 - 0.984i)T^{2} \) |
| 97 | \( 1 + (-0.592 - 0.342i)T + (0.5 + 0.866i)T^{2} \) |
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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
−8.860146655925355170248592309855, −7.895062789727798262290404248871, −7.42401083720784449586961797723, −6.96184022539223181041431311353, −5.80355487181440394322294798213, −4.98451437110822605140800065789, −4.22629597189566037154016820892, −3.15941871946395385587645593493, −2.58523167631480230657214277895, −0.909547827187092365498594945939,
1.12790065694083552813510880975, 2.20037348985892043440343801775, 3.61616688755070413367866520362, 4.35674700807090777549910171188, 4.70431399925468531602213810313, 6.07712259022525852271068700115, 6.63870569572979774601107235902, 7.66120980984226395953395666930, 7.994961974767740473577445334497, 9.085780639321298968520835541473