| L(s) = 1 | − 1.41·2-s − 4.30·3-s + 2.00·4-s + 6.09·6-s − 1.47i·7-s − 2.82·8-s + 9.55·9-s + 6.04i·11-s − 8.61·12-s + 5.21·13-s + 2.08i·14-s + 4.00·16-s + 15.7i·17-s − 13.5·18-s − 4.82i·19-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.43·3-s + 0.500·4-s + 1.01·6-s − 0.210i·7-s − 0.353·8-s + 1.06·9-s + 0.549i·11-s − 0.717·12-s + 0.401·13-s + 0.149i·14-s + 0.250·16-s + 0.923i·17-s − 0.750·18-s − 0.254i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.112 - 0.993i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.112 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4774900980\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4774900980\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41T \) |
| 5 | \( 1 \) |
| 23 | \( 1 + (-2.58 + 22.8i)T \) |
| good | 3 | \( 1 + 4.30T + 9T^{2} \) |
| 7 | \( 1 + 1.47iT - 49T^{2} \) |
| 11 | \( 1 - 6.04iT - 121T^{2} \) |
| 13 | \( 1 - 5.21T + 169T^{2} \) |
| 17 | \( 1 - 15.7iT - 289T^{2} \) |
| 19 | \( 1 + 4.82iT - 361T^{2} \) |
| 29 | \( 1 + 23.4T + 841T^{2} \) |
| 31 | \( 1 + 20.4T + 961T^{2} \) |
| 37 | \( 1 - 15.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 20.3T + 1.68e3T^{2} \) |
| 43 | \( 1 + 38.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 13.8T + 2.20e3T^{2} \) |
| 53 | \( 1 + 38.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 33.5T + 3.48e3T^{2} \) |
| 61 | \( 1 + 100. iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 32.4iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 24.1T + 5.04e3T^{2} \) |
| 73 | \( 1 + 15.1T + 5.32e3T^{2} \) |
| 79 | \( 1 - 11.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 44.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 111. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 154. iT - 9.40e3T^{2} \) |
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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
−9.976016818629063420569847277585, −9.071707479474133037724222758018, −8.147718100338350230960707800864, −7.16805234189508262130032102782, −6.48321677782436241958038164795, −5.74762589612567431979625394390, −4.82212679964387436484902214124, −3.74290844550447033154607667956, −2.08724458129326578986036600153, −0.842864888827387019100135328038,
0.31826788554438434166720692718, 1.44614102273328544603029111642, 2.99491527329880359531561186152, 4.32543136364643765651605018177, 5.66786031500939400042340105162, 5.77499800604803091457823976387, 6.97470089281643375262837855945, 7.60156795267654325919718189852, 8.766704422746539690031996090612, 9.446803571142536040461498857694
