| L(s) = 1 | − 3-s + 3.06·5-s − 2.06·7-s + 9-s − 6.45·11-s + 13-s − 3.06·15-s + 1.38·17-s + 2.06·21-s + 3.06·23-s + 4.38·25-s − 27-s − 3.51·29-s + 9.45·31-s + 6.45·33-s − 6.32·35-s − 2.38·37-s − 39-s + 10.1·41-s + 6.06·43-s + 3.06·45-s − 6·47-s − 2.73·49-s − 1.38·51-s − 10.5·53-s − 19.7·55-s − 11.1·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.37·5-s − 0.780·7-s + 0.333·9-s − 1.94·11-s + 0.277·13-s − 0.791·15-s + 0.336·17-s + 0.450·21-s + 0.638·23-s + 0.877·25-s − 0.192·27-s − 0.653·29-s + 1.69·31-s + 1.12·33-s − 1.06·35-s − 0.392·37-s − 0.160·39-s + 1.58·41-s + 0.924·43-s + 0.456·45-s − 0.875·47-s − 0.391·49-s − 0.194·51-s − 1.45·53-s − 2.66·55-s − 1.45·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.558287791\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.558287791\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 - 3.06T + 5T^{2} \) |
| 7 | \( 1 + 2.06T + 7T^{2} \) |
| 11 | \( 1 + 6.45T + 11T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 - 1.38T + 17T^{2} \) |
| 23 | \( 1 - 3.06T + 23T^{2} \) |
| 29 | \( 1 + 3.51T + 29T^{2} \) |
| 31 | \( 1 - 9.45T + 31T^{2} \) |
| 37 | \( 1 + 2.38T + 37T^{2} \) |
| 41 | \( 1 - 10.1T + 41T^{2} \) |
| 43 | \( 1 - 6.06T + 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 + 10.5T + 53T^{2} \) |
| 59 | \( 1 + 11.1T + 59T^{2} \) |
| 61 | \( 1 - 5.12T + 61T^{2} \) |
| 67 | \( 1 - 3.45T + 67T^{2} \) |
| 71 | \( 1 - 6.73T + 71T^{2} \) |
| 73 | \( 1 + 9.12T + 73T^{2} \) |
| 79 | \( 1 + 1.58T + 79T^{2} \) |
| 83 | \( 1 + 17.6T + 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 - 6.73T + 97T^{2} \) |
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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
−7.69970872481682439332638157005, −6.97955655948639616572204810603, −6.03462825957970128834091706652, −5.94826416969096767814560529695, −5.11224477473601325332915738646, −4.54398268716957126516695490866, −3.19187081965832506254021215981, −2.68562951122999337768648475962, −1.77930265050909609774735114436, −0.61307013884533657001285808745,
0.61307013884533657001285808745, 1.77930265050909609774735114436, 2.68562951122999337768648475962, 3.19187081965832506254021215981, 4.54398268716957126516695490866, 5.11224477473601325332915738646, 5.94826416969096767814560529695, 6.03462825957970128834091706652, 6.97955655948639616572204810603, 7.69970872481682439332638157005
