L(s) = 1 | + (0.167 + 1.40i)2-s + 2.41i·3-s + (−1.94 + 0.470i)4-s + i·5-s + (−3.39 + 0.404i)6-s − 7-s + (−0.985 − 2.65i)8-s − 2.82·9-s + (−1.40 + 0.167i)10-s − 2.49i·11-s + (−1.13 − 4.69i)12-s + 6.97i·13-s + (−0.167 − 1.40i)14-s − 2.41·15-s + (3.55 − 1.82i)16-s + 4.03·17-s + ⋯ |
L(s) = 1 | + (0.118 + 0.992i)2-s + 1.39i·3-s + (−0.971 + 0.235i)4-s + 0.447i·5-s + (−1.38 + 0.165i)6-s − 0.377·7-s + (−0.348 − 0.937i)8-s − 0.942·9-s + (−0.444 + 0.0529i)10-s − 0.753i·11-s + (−0.327 − 1.35i)12-s + 1.93i·13-s + (−0.0447 − 0.375i)14-s − 0.623·15-s + (0.889 − 0.457i)16-s + 0.977·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.937 + 0.348i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.937 + 0.348i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.188169 - 1.04577i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.188169 - 1.04577i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.167 - 1.40i)T \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + T \) |
good | 3 | \( 1 - 2.41iT - 3T^{2} \) |
| 11 | \( 1 + 2.49iT - 11T^{2} \) |
| 13 | \( 1 - 6.97iT - 13T^{2} \) |
| 17 | \( 1 - 4.03T + 17T^{2} \) |
| 19 | \( 1 + 4.64iT - 19T^{2} \) |
| 23 | \( 1 + 2.35T + 23T^{2} \) |
| 29 | \( 1 - 9.33iT - 29T^{2} \) |
| 31 | \( 1 + 0.315T + 31T^{2} \) |
| 37 | \( 1 + 7.30iT - 37T^{2} \) |
| 41 | \( 1 + 2.64T + 41T^{2} \) |
| 43 | \( 1 - 1.68iT - 43T^{2} \) |
| 47 | \( 1 - 4.18T + 47T^{2} \) |
| 53 | \( 1 - 9.11iT - 53T^{2} \) |
| 59 | \( 1 - 11.7iT - 59T^{2} \) |
| 61 | \( 1 + 5.03iT - 61T^{2} \) |
| 67 | \( 1 + 5.11iT - 67T^{2} \) |
| 71 | \( 1 - 7.89T + 71T^{2} \) |
| 73 | \( 1 - 9.89T + 73T^{2} \) |
| 79 | \( 1 - 2.99T + 79T^{2} \) |
| 83 | \( 1 + 4.11iT - 83T^{2} \) |
| 89 | \( 1 - 12.5T + 89T^{2} \) |
| 97 | \( 1 - 17.7T + 97T^{2} \) |
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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
−12.40631229419255699688736476723, −11.23478093344205288030169458212, −10.31124569998540946681221018542, −9.249907363364219972900548909257, −8.925636646107868657138526471982, −7.36725370882089239812187835998, −6.40794153546061316515575342830, −5.28358459059158632340999585719, −4.22970037971459393661806953558, −3.33386684072321117868862346840,
0.822670326486147477649656091538, 2.20692449009271089806317985302, 3.58167701314939812941250014243, 5.24350371540809625595153757818, 6.19280512107900000351368381566, 7.83497473567751733946190790064, 8.143420939619945551694450649427, 9.776097802878067840181601709135, 10.28770695815557750404206822322, 11.81740020678353690434733012099