| L(s) = 1 | − i·3-s − 0.913·7-s − 9-s + 3.58i·11-s − 0.913i·13-s − 3.58·17-s − 4i·19-s + 0.913i·21-s + i·27-s + 7.84i·29-s + 5.29·31-s + 3.58·33-s − 7.84i·37-s − 0.913·39-s + 6·41-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.345·7-s − 0.333·9-s + 1.08i·11-s − 0.253i·13-s − 0.868·17-s − 0.917i·19-s + 0.199i·21-s + 0.192i·27-s + 1.45i·29-s + 0.950·31-s + 0.623·33-s − 1.28i·37-s − 0.146·39-s + 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.426213558\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.426213558\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 + 0.913T + 7T^{2} \) |
| 11 | \( 1 - 3.58iT - 11T^{2} \) |
| 13 | \( 1 + 0.913iT - 13T^{2} \) |
| 17 | \( 1 + 3.58T + 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 7.84iT - 29T^{2} \) |
| 31 | \( 1 - 5.29T + 31T^{2} \) |
| 37 | \( 1 + 7.84iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 7.16iT - 43T^{2} \) |
| 47 | \( 1 - 6.92T + 47T^{2} \) |
| 53 | \( 1 + 2.55iT - 53T^{2} \) |
| 59 | \( 1 + 7.58iT - 59T^{2} \) |
| 61 | \( 1 + 10.5iT - 61T^{2} \) |
| 67 | \( 1 + 15.1iT - 67T^{2} \) |
| 71 | \( 1 - 6.92T + 71T^{2} \) |
| 73 | \( 1 + 12T + 73T^{2} \) |
| 79 | \( 1 - 5.29T + 79T^{2} \) |
| 83 | \( 1 + 11.1iT - 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 - 7.16T + 97T^{2} \) |
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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
−7.955423836709172440383004668700, −7.39658898309949356728714903907, −6.67504140475983957943324966790, −6.21150637096262889254351971339, −5.06986174346021023996917341951, −4.57285968633510381385347279929, −3.46349215176443719226676253090, −2.56180492154451444702611786796, −1.78373338528382709375758037263, −0.48613934769366616177649960033,
0.877150158737686290653930227651, 2.31454217451708387485838062418, 3.11222578320205227274049043018, 4.02173834234305425674313509513, 4.53764703516350250118203657894, 5.72446831005512941127278530772, 6.04719826893803875318370522484, 6.91294924445921538551029124659, 7.84777772453426286712972802396, 8.568561438449202384348402944806
