| L(s) = 1 | + 2.23·5-s + 0.763i·7-s − 5.70i·23-s + 5.00·25-s + 6·29-s + 1.70i·35-s + 4.47·41-s + 11.2i·43-s − 13.7i·47-s + 6.41·49-s + 13.4·61-s + 8.18i·67-s + 17.7i·83-s + 6·89-s − 18·101-s + ⋯ |
| L(s) = 1 | + 0.999·5-s + 0.288i·7-s − 1.19i·23-s + 1.00·25-s + 1.11·29-s + 0.288i·35-s + 0.698·41-s + 1.71i·43-s − 1.99i·47-s + 0.916·49-s + 1.71·61-s + 0.999i·67-s + 1.94i·83-s + 0.635·89-s − 1.79·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.315405008\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.315405008\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 2.23T \) |
| good | 7 | \( 1 - 0.763iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 5.70iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 4.47T + 41T^{2} \) |
| 43 | \( 1 - 11.2iT - 43T^{2} \) |
| 47 | \( 1 + 13.7iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 13.4T + 61T^{2} \) |
| 67 | \( 1 - 8.18iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 17.7iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−8.691518291641107442958887433071, −8.271380049059628321332090999019, −7.06713571975005963752516038501, −6.48702402891825224132569700430, −5.72101687116164328488515604481, −4.99914656660973082562287188235, −4.11267735377046804860645337326, −2.85557421131888580918419955966, −2.20586783032921113030241401501, −0.954832313395770877247254012070,
0.993273927838001415845283597818, 2.06609336450487304540369509053, 3.01226164677184864662559904674, 4.04156076307849215339144300755, 5.01173497006916718995512074122, 5.71397999971242062779432678025, 6.45041939670361850725115845541, 7.21880635147988780352206291818, 7.996826343562514522282134623264, 8.956655899523247383029630049873
