| L(s) = 1 | + 2.04·5-s + 7.79·7-s + 4.09·11-s − 6.69i·13-s − 19.3i·17-s − 1.79i·19-s − 14.1i·23-s − 20.7·25-s + 28.4·29-s − 20.2·31-s + 15.9·35-s − 41.5i·37-s − 4.94i·41-s − 75.1i·43-s − 13.5i·47-s + ⋯ |
| L(s) = 1 | + 0.409·5-s + 1.11·7-s + 0.372·11-s − 0.515i·13-s − 1.13i·17-s − 0.0946i·19-s − 0.614i·23-s − 0.831·25-s + 0.982·29-s − 0.651·31-s + 0.456·35-s − 1.12i·37-s − 0.120i·41-s − 1.74i·43-s − 0.288i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.169 + 0.985i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.169 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.303606026\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.303606026\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 2.04T + 25T^{2} \) |
| 7 | \( 1 - 7.79T + 49T^{2} \) |
| 11 | \( 1 - 4.09T + 121T^{2} \) |
| 13 | \( 1 + 6.69iT - 169T^{2} \) |
| 17 | \( 1 + 19.3iT - 289T^{2} \) |
| 19 | \( 1 + 1.79iT - 361T^{2} \) |
| 23 | \( 1 + 14.1iT - 529T^{2} \) |
| 29 | \( 1 - 28.4T + 841T^{2} \) |
| 31 | \( 1 + 20.2T + 961T^{2} \) |
| 37 | \( 1 + 41.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 4.94iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 75.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 13.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 20.8T + 2.80e3T^{2} \) |
| 59 | \( 1 + 54.8T + 3.48e3T^{2} \) |
| 61 | \( 1 - 89.1iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 37.7iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 117. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 48.4T + 5.32e3T^{2} \) |
| 79 | \( 1 + 92.6T + 6.24e3T^{2} \) |
| 83 | \( 1 - 158.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 121. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 167.T + 9.40e3T^{2} \) |
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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
−8.753632826533289175588495706938, −7.81524642726934881386658503778, −7.25521525124421562876270836709, −6.26002918696837613414985138750, −5.40068941940463921564092019133, −4.77933394790707348425994317370, −3.82169964955370378033525820823, −2.63020337262996548765053383897, −1.74804675717276220008301483279, −0.55076975769308746395098553299,
1.35124651707572352660078272690, 1.91970022867571714731343811768, 3.25236999563271432312928163855, 4.29862463186706022327010440251, 4.93700370626928677592707404017, 5.96873511521135567231810161672, 6.51514237742815344219543654029, 7.65702758804568557242996677993, 8.161024002023642796773493965608, 8.964705468254601376323301072082
