L(s) = 1 | − 2i·5-s + 7-s + 4i·11-s − 2i·13-s + 2·17-s − 6·23-s + 25-s − 2i·29-s + 4·31-s − 2i·35-s − 4i·37-s + 10·41-s − 2i·43-s + 8·47-s + 49-s + ⋯ |
L(s) = 1 | − 0.894i·5-s + 0.377·7-s + 1.20i·11-s − 0.554i·13-s + 0.485·17-s − 1.25·23-s + 0.200·25-s − 0.371i·29-s + 0.718·31-s − 0.338i·35-s − 0.657i·37-s + 1.56·41-s − 0.304i·43-s + 1.16·47-s + 0.142·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.958212049\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.958212049\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 + 2iT - 5T^{2} \) |
| 11 | \( 1 - 4iT - 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 + 2iT - 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 + 2iT - 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 4iT - 59T^{2} \) |
| 61 | \( 1 + 10iT - 61T^{2} \) |
| 67 | \( 1 + 2iT - 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−8.200686335427796865503521379987, −7.78688872305237803604980339164, −7.01686402147851736568768611083, −5.96576646339378720692679316871, −5.35845568595979807839408016799, −4.50793801986987985479109833042, −4.02328660278433705918852372644, −2.69463069805405947864151000994, −1.78423328056731611659488669630, −0.70999647413327453779454816643,
0.950956034959552966560564407694, 2.22795057727428517874093116653, 3.06740082358320647091756732469, 3.84443787968040537033968804552, 4.73007558741843598847849267318, 5.80251375175720343170288047653, 6.23009748953773225726142378952, 7.08969589350899637640021816754, 7.80381142093737616370992922728, 8.471838962770586415948708454519