L(s) = 1 | + (−0.707 − 0.707i)2-s − 0.999i·4-s + (2 − 2i)7-s + (−2.12 + 2.12i)8-s − 2.82i·11-s + (−1 − i)13-s − 2.82·14-s + 1.00·16-s + (−2.82 − 2.82i)17-s + (−2.00 + 2.00i)22-s + (2.82 − 2.82i)23-s + 1.41i·26-s + (−1.99 − 1.99i)28-s + 4.24·29-s − 4·31-s + (3.53 + 3.53i)32-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s − 0.499i·4-s + (0.755 − 0.755i)7-s + (−0.750 + 0.750i)8-s − 0.852i·11-s + (−0.277 − 0.277i)13-s − 0.755·14-s + 0.250·16-s + (−0.685 − 0.685i)17-s + (−0.426 + 0.426i)22-s + (0.589 − 0.589i)23-s + 0.277i·26-s + (−0.377 − 0.377i)28-s + 0.787·29-s − 0.718·31-s + (0.624 + 0.624i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.391 + 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.391 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.502559 - 0.759545i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.502559 - 0.759545i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (0.707 + 0.707i)T + 2iT^{2} \) |
| 7 | \( 1 + (-2 + 2i)T - 7iT^{2} \) |
| 11 | \( 1 + 2.82iT - 11T^{2} \) |
| 13 | \( 1 + (1 + i)T + 13iT^{2} \) |
| 17 | \( 1 + (2.82 + 2.82i)T + 17iT^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + (-2.82 + 2.82i)T - 23iT^{2} \) |
| 29 | \( 1 - 4.24T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + (1 - i)T - 37iT^{2} \) |
| 41 | \( 1 - 1.41iT - 41T^{2} \) |
| 43 | \( 1 + (-8 - 8i)T + 43iT^{2} \) |
| 47 | \( 1 + (-5.65 - 5.65i)T + 47iT^{2} \) |
| 53 | \( 1 + (-2.82 + 2.82i)T - 53iT^{2} \) |
| 59 | \( 1 - 8.48T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 + (4 - 4i)T - 67iT^{2} \) |
| 71 | \( 1 - 5.65iT - 71T^{2} \) |
| 73 | \( 1 + (1 + i)T + 73iT^{2} \) |
| 79 | \( 1 - 12iT - 79T^{2} \) |
| 83 | \( 1 + (-2.82 + 2.82i)T - 83iT^{2} \) |
| 89 | \( 1 + 12.7T + 89T^{2} \) |
| 97 | \( 1 + (-11 + 11i)T - 97iT^{2} \) |
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
−11.46959344371491948737042516898, −11.00891156892627751983405687857, −10.16410884146628574775607099986, −9.087039964230170588001187478423, −8.207357606127557863819765877889, −6.95251237299363432805196377715, −5.63791021885016944402708361816, −4.49273148215298350663438184775, −2.67939316892579796084197521046, −0.941842629029834829276222122057,
2.25848312992364474183984224563, 4.01941594741982611705807245109, 5.36123438529372113779581079301, 6.74815821322440349680070626144, 7.60199167602809397153443426663, 8.646272972011338520076968077525, 9.251053975480417444355696257082, 10.54996192158307526478698564559, 11.77595465101173436169065162725, 12.37564912344075466035146942045