L(s) = 1 | − i·2-s + 2.61i·3-s − 4-s − 2.61i·5-s + 2.61·6-s + i·8-s − 3.82·9-s − 2.61·10-s + (1.41 + 3i)11-s − 2.61i·12-s − 5.54·13-s + 6.82·15-s + 16-s + 3.82i·18-s − 2.29·19-s + 2.61i·20-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + 1.50i·3-s − 0.5·4-s − 1.16i·5-s + 1.06·6-s + 0.353i·8-s − 1.27·9-s − 0.826·10-s + (0.426 + 0.904i)11-s − 0.754i·12-s − 1.53·13-s + 1.76·15-s + 0.250·16-s + 0.902i·18-s − 0.526·19-s + 0.584i·20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.562 - 0.826i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.562 - 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6866117141\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6866117141\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (-1.41 - 3i)T \) |
good | 3 | \( 1 - 2.61iT - 3T^{2} \) |
| 5 | \( 1 + 2.61iT - 5T^{2} \) |
| 13 | \( 1 + 5.54T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 2.29T + 19T^{2} \) |
| 23 | \( 1 + 2.24T + 23T^{2} \) |
| 29 | \( 1 - 10.2iT - 29T^{2} \) |
| 31 | \( 1 - 7.07iT - 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + 11.0T + 41T^{2} \) |
| 43 | \( 1 + 4.24iT - 43T^{2} \) |
| 47 | \( 1 - 6.62iT - 47T^{2} \) |
| 53 | \( 1 + 12.4T + 53T^{2} \) |
| 59 | \( 1 + 12.1iT - 59T^{2} \) |
| 61 | \( 1 + 2.29T + 61T^{2} \) |
| 67 | \( 1 - 0.343T + 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 + 6.49T + 73T^{2} \) |
| 79 | \( 1 - 8.48iT - 79T^{2} \) |
| 83 | \( 1 - 0.951T + 83T^{2} \) |
| 89 | \( 1 - 12.3iT - 89T^{2} \) |
| 97 | \( 1 + 5.09iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.09783301332224894388521381631, −9.408524716883970127370532366634, −9.006465011152017465104755559095, −8.066204050838795834111403162205, −6.79801443612339120887983178314, −5.10375210211732710243520898418, −4.95027399959015322639988802610, −4.17402483941321831581903643164, −3.11029106768976452479271521201, −1.68587529569624291891029104163,
0.29429501799943452744540486802, 2.08814824533555004383526653048, 3.00655166299228507508366607090, 4.40588201217524698434149434653, 5.89823712457578864445179315484, 6.32682382832000677485711680144, 7.09581136443754980738629573899, 7.71357956071731183825282770869, 8.318178414438514577018026518827, 9.533451367146359238837902499003