| 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 | \( 1 + (-11 - 11i)T + 97iT^{2} \) |
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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
−12.37564912344075466035146942045, −11.77595465101173436169065162725, −10.54996192158307526478698564559, −9.251053975480417444355696257082, −8.646272972011338520076968077525, −7.60199167602809397153443426663, −6.74815821322440349680070626144, −5.36123438529372113779581079301, −4.01941594741982611705807245109, −2.25848312992364474183984224563,
0.941842629029834829276222122057, 2.67939316892579796084197521046, 4.49273148215298350663438184775, 5.63791021885016944402708361816, 6.95251237299363432805196377715, 8.207357606127557863819765877889, 9.087039964230170588001187478423, 10.16410884146628574775607099986, 11.00891156892627751983405687857, 11.46959344371491948737042516898
