| L(s) = 1 | − i·2-s − 1.61i·3-s − 4-s − 1.61·6-s + 3.23i·7-s + i·8-s + 0.381·9-s − 1.38·11-s + 1.61i·12-s − 2.85i·13-s + 3.23·14-s + 16-s + 4.47i·17-s − 0.381i·18-s − 4.47·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.934i·3-s − 0.5·4-s − 0.660·6-s + 1.22i·7-s + 0.353i·8-s + 0.127·9-s − 0.416·11-s + 0.467i·12-s − 0.791i·13-s + 0.864·14-s + 0.250·16-s + 1.08i·17-s − 0.0900i·18-s − 1.02·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.617888744\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.617888744\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 37 | \( 1 - iT \) |
| good | 3 | \( 1 + 1.61iT - 3T^{2} \) |
| 7 | \( 1 - 3.23iT - 7T^{2} \) |
| 11 | \( 1 + 1.38T + 11T^{2} \) |
| 13 | \( 1 + 2.85iT - 13T^{2} \) |
| 17 | \( 1 - 4.47iT - 17T^{2} \) |
| 19 | \( 1 + 4.47T + 19T^{2} \) |
| 23 | \( 1 - 2.85iT - 23T^{2} \) |
| 29 | \( 1 - 9.32T + 29T^{2} \) |
| 31 | \( 1 - 7.38T + 31T^{2} \) |
| 41 | \( 1 - 9.61T + 41T^{2} \) |
| 43 | \( 1 + 5.23iT - 43T^{2} \) |
| 47 | \( 1 - 1.23iT - 47T^{2} \) |
| 53 | \( 1 - 0.472iT - 53T^{2} \) |
| 59 | \( 1 - 4.76T + 59T^{2} \) |
| 61 | \( 1 - 10.6T + 61T^{2} \) |
| 67 | \( 1 + 1.09iT - 67T^{2} \) |
| 71 | \( 1 - 2.94T + 71T^{2} \) |
| 73 | \( 1 - 7.09iT - 73T^{2} \) |
| 79 | \( 1 - 8.56T + 79T^{2} \) |
| 83 | \( 1 + 14.4iT - 83T^{2} \) |
| 89 | \( 1 - 1.52T + 89T^{2} \) |
| 97 | \( 1 - 0.472iT - 97T^{2} \) |
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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
−8.978345255654167288384850531620, −8.297478271473894408222395842848, −7.82788677168211020512433610675, −6.57744267414963041529921279745, −5.97776776906667003131956065280, −5.05796428760295281255131083536, −4.00736918242936213849643868015, −2.72460168941114461722244787920, −2.16333881942668268026299254339, −0.949302567645226552516597035641,
0.819968824644733269039027355521, 2.67352325812568239049949758056, 3.99805437818644951159728598564, 4.45134085633356656992648928664, 5.07353595753778479334091438657, 6.42925279594925202743322124885, 6.89968418478390997393000534194, 7.78587326954008126495215528353, 8.579170783343341143170954057257, 9.449546003843632828412422409069
