L(s) = 1 | + (2.95 − 0.5i)3-s + 8i·7-s + (8.5 − 2.95i)9-s − 17.7i·11-s + 2i·13-s − 17.7·17-s + 11·19-s + (4 + 23.6i)21-s + 35.4·23-s + (23.6 − 13i)27-s − 35.4i·29-s + 46·31-s + (−8.87 − 52.5i)33-s + 16i·37-s + (1 + 5.91i)39-s + ⋯ |
L(s) = 1 | + (0.986 − 0.166i)3-s + 1.14i·7-s + (0.944 − 0.328i)9-s − 1.61i·11-s + 0.153i·13-s − 1.04·17-s + 0.578·19-s + (0.190 + 1.12i)21-s + 1.54·23-s + (0.876 − 0.481i)27-s − 1.22i·29-s + 1.48·31-s + (−0.268 − 1.59i)33-s + 0.432i·37-s + (0.0256 + 0.151i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 + 0.291i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.956 + 0.291i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.989012073\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.989012073\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.95 + 0.5i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 8iT - 49T^{2} \) |
| 11 | \( 1 + 17.7iT - 121T^{2} \) |
| 13 | \( 1 - 2iT - 169T^{2} \) |
| 17 | \( 1 + 17.7T + 289T^{2} \) |
| 19 | \( 1 - 11T + 361T^{2} \) |
| 23 | \( 1 - 35.4T + 529T^{2} \) |
| 29 | \( 1 + 35.4iT - 841T^{2} \) |
| 31 | \( 1 - 46T + 961T^{2} \) |
| 37 | \( 1 - 16iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 53.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 62iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 35.4T + 2.20e3T^{2} \) |
| 53 | \( 1 - 35.4T + 2.80e3T^{2} \) |
| 59 | \( 1 + 70.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 16T + 3.72e3T^{2} \) |
| 67 | \( 1 - 113iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 106. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 101iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 68T + 6.24e3T^{2} \) |
| 83 | \( 1 + 17.7T + 6.88e3T^{2} \) |
| 89 | \( 1 + 53.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 22iT - 9.40e3T^{2} \) |
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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
−9.259389935064725471178254539043, −8.600352248284479043837990880166, −8.271412926576655777603626853969, −7.05224674254208567949547278580, −6.24281035635325545081514209020, −5.31793937448084755863499573908, −4.15915230797381405777562245236, −3.00394983330609464481894035068, −2.46404992187928207551284584523, −0.953048063067222380272876681557,
1.13954497021868205143885352021, 2.33134638063580327888358275835, 3.40341900252394626303738754827, 4.43437440465103462244890480756, 4.90239361223595425291033578893, 6.67436252943759234519554843560, 7.23501853809427025015583804779, 7.80173064603054231588348678116, 8.984613580647267597263818057646, 9.433950102306105452774014026116