| L(s) = 1 | + i·3-s − i·5-s − 2·7-s − 9-s − 2i·11-s + 2i·13-s + 15-s − 2·17-s − 2i·19-s − 2i·21-s − 2·23-s − 25-s − i·27-s + 6i·29-s + 4·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.447i·5-s − 0.755·7-s − 0.333·9-s − 0.603i·11-s + 0.554i·13-s + 0.258·15-s − 0.485·17-s − 0.458i·19-s − 0.436i·21-s − 0.417·23-s − 0.200·25-s − 0.192i·27-s + 1.11i·29-s + 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.383162384\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.383162384\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + iT \) |
| good | 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 2iT - 19T^{2} \) |
| 23 | \( 1 + 2T + 23T^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 - 2T + 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 2iT - 59T^{2} \) |
| 61 | \( 1 - 10iT - 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 - 16T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−8.774952585729965456129835243451, −7.993912482476684355209898096027, −6.99942461829639485527240670252, −6.32539431110793658148666083920, −5.58083512635256841301766302033, −4.73637392265189121320659849815, −4.00343606549441224667644928499, −3.19988952236857052127572822872, −2.24930454546685715341295803470, −0.78859289201617975783698704764,
0.55929500020439937515470740784, 1.99369228606659514884963840121, 2.74643693783960879489484546063, 3.65577386419758656116920396697, 4.55047325233072525417247384495, 5.62232690846816836210592116059, 6.36100173357021849664235289123, 6.76742743840112987074920504934, 7.83256393604755001192482682697, 8.033509809642357855705635274075
