| L(s) = 1 | + 3-s − 3.56·5-s − 1.56·7-s + 9-s − 1.56·11-s − 5.12·13-s − 3.56·15-s + 7.56·17-s − 19-s − 1.56·21-s + 7.68·25-s + 27-s + 5.12·29-s + 7.12·31-s − 1.56·33-s + 5.56·35-s + 1.12·37-s − 5.12·39-s − 1.12·41-s + 12.6·43-s − 3.56·45-s + 5.56·47-s − 4.56·49-s + 7.56·51-s + 8.24·53-s + 5.56·55-s − 57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.59·5-s − 0.590·7-s + 0.333·9-s − 0.470·11-s − 1.42·13-s − 0.919·15-s + 1.83·17-s − 0.229·19-s − 0.340·21-s + 1.53·25-s + 0.192·27-s + 0.951·29-s + 1.27·31-s − 0.271·33-s + 0.940·35-s + 0.184·37-s − 0.820·39-s − 0.175·41-s + 1.93·43-s − 0.530·45-s + 0.811·47-s − 0.651·49-s + 1.05·51-s + 1.13·53-s + 0.749·55-s − 0.132·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1824 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1824 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.235412477\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.235412477\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| good | 5 | \( 1 + 3.56T + 5T^{2} \) |
| 7 | \( 1 + 1.56T + 7T^{2} \) |
| 11 | \( 1 + 1.56T + 11T^{2} \) |
| 13 | \( 1 + 5.12T + 13T^{2} \) |
| 17 | \( 1 - 7.56T + 17T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 5.12T + 29T^{2} \) |
| 31 | \( 1 - 7.12T + 31T^{2} \) |
| 37 | \( 1 - 1.12T + 37T^{2} \) |
| 41 | \( 1 + 1.12T + 41T^{2} \) |
| 43 | \( 1 - 12.6T + 43T^{2} \) |
| 47 | \( 1 - 5.56T + 47T^{2} \) |
| 53 | \( 1 - 8.24T + 53T^{2} \) |
| 59 | \( 1 - 0.876T + 59T^{2} \) |
| 61 | \( 1 + 7.56T + 61T^{2} \) |
| 67 | \( 1 + 13.3T + 67T^{2} \) |
| 71 | \( 1 - 4.87T + 71T^{2} \) |
| 73 | \( 1 + 14.6T + 73T^{2} \) |
| 79 | \( 1 - 7.12T + 79T^{2} \) |
| 83 | \( 1 - 10.2T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 - 13.1T + 97T^{2} \) |
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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
−9.196318367665296717235505980674, −8.302247943424898866820183326681, −7.55896167382606374727116314133, −7.38343552883972513989297587542, −6.14484580834483961262068321550, −4.94951514056107150074606520218, −4.21309971657547831765520075487, −3.25042217833407368542448381221, −2.64475642409332810402103952953, −0.71876446413246172913360057893,
0.71876446413246172913360057893, 2.64475642409332810402103952953, 3.25042217833407368542448381221, 4.21309971657547831765520075487, 4.94951514056107150074606520218, 6.14484580834483961262068321550, 7.38343552883972513989297587542, 7.55896167382606374727116314133, 8.302247943424898866820183326681, 9.196318367665296717235505980674
