| L(s) = 1 | − 3-s − 1.56·5-s − 1.56·7-s + 9-s + 5.56·11-s − 3.12·13-s + 1.56·15-s + 6.68·17-s + 1.56·21-s + 9.12·23-s − 2.56·25-s − 27-s + 8.24·29-s + 2·31-s − 5.56·33-s + 2.43·35-s + 8·37-s + 3.12·39-s + 5.12·41-s − 1.56·43-s − 1.56·45-s − 6.68·47-s − 4.56·49-s − 6.68·51-s − 4.24·53-s − 8.68·55-s − 12·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.698·5-s − 0.590·7-s + 0.333·9-s + 1.67·11-s − 0.866·13-s + 0.403·15-s + 1.62·17-s + 0.340·21-s + 1.90·23-s − 0.512·25-s − 0.192·27-s + 1.53·29-s + 0.359·31-s − 0.968·33-s + 0.412·35-s + 1.31·37-s + 0.500·39-s + 0.800·41-s − 0.238·43-s − 0.232·45-s − 0.975·47-s − 0.651·49-s − 0.936·51-s − 0.583·53-s − 1.17·55-s − 1.56·59-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
| \(L(1)\) |
\(\approx\) |
\(1.559264637\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.559264637\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.56T + 5T^{2} \) |
| 7 | \( 1 + 1.56T + 7T^{2} \) |
| 11 | \( 1 - 5.56T + 11T^{2} \) |
| 13 | \( 1 + 3.12T + 13T^{2} \) |
| 17 | \( 1 - 6.68T + 17T^{2} \) |
| 23 | \( 1 - 9.12T + 23T^{2} \) |
| 29 | \( 1 - 8.24T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 - 5.12T + 41T^{2} \) |
| 43 | \( 1 + 1.56T + 43T^{2} \) |
| 47 | \( 1 + 6.68T + 47T^{2} \) |
| 53 | \( 1 + 4.24T + 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 - 6.68T + 61T^{2} \) |
| 67 | \( 1 + 6.24T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 16.9T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 4.24T + 97T^{2} \) |
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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
−7.56624374017369173360267477604, −7.11339746931643261577948371825, −6.35752585674426742731213367594, −5.89013037580881614207595070769, −4.78612574162563156363889971623, −4.42560714919031269394240036267, −3.38885374221572810056388763432, −2.94958505022704096011496244553, −1.42106695766733803428492063632, −0.69512806200963122090633853821,
0.69512806200963122090633853821, 1.42106695766733803428492063632, 2.94958505022704096011496244553, 3.38885374221572810056388763432, 4.42560714919031269394240036267, 4.78612574162563156363889971623, 5.89013037580881614207595070769, 6.35752585674426742731213367594, 7.11339746931643261577948371825, 7.56624374017369173360267477604
