| L(s) = 1 | + 2-s − 3-s + 4-s + 1.69·5-s − 6-s − 7-s + 8-s + 9-s + 1.69·10-s + 3.15·11-s − 12-s − 14-s − 1.69·15-s + 16-s + 7.74·17-s + 18-s + 3.15·19-s + 1.69·20-s + 21-s + 3.15·22-s + 7.89·23-s − 24-s − 2.13·25-s − 27-s − 28-s − 4.96·29-s − 1.69·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.756·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.535·10-s + 0.952·11-s − 0.288·12-s − 0.267·14-s − 0.436·15-s + 0.250·16-s + 1.87·17-s + 0.235·18-s + 0.724·19-s + 0.378·20-s + 0.218·21-s + 0.673·22-s + 1.64·23-s − 0.204·24-s − 0.427·25-s − 0.192·27-s − 0.188·28-s − 0.921·29-s − 0.308·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.706409094\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.706409094\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 1.69T + 5T^{2} \) |
| 11 | \( 1 - 3.15T + 11T^{2} \) |
| 17 | \( 1 - 7.74T + 17T^{2} \) |
| 19 | \( 1 - 3.15T + 19T^{2} \) |
| 23 | \( 1 - 7.89T + 23T^{2} \) |
| 29 | \( 1 + 4.96T + 29T^{2} \) |
| 31 | \( 1 - 4.82T + 31T^{2} \) |
| 37 | \( 1 + 0.185T + 37T^{2} \) |
| 41 | \( 1 - 0.978T + 41T^{2} \) |
| 43 | \( 1 + 8.19T + 43T^{2} \) |
| 47 | \( 1 - 2.78T + 47T^{2} \) |
| 53 | \( 1 + 0.652T + 53T^{2} \) |
| 59 | \( 1 + 4.80T + 59T^{2} \) |
| 61 | \( 1 - 3.21T + 61T^{2} \) |
| 67 | \( 1 + 4.52T + 67T^{2} \) |
| 71 | \( 1 + 1.20T + 71T^{2} \) |
| 73 | \( 1 + 14.1T + 73T^{2} \) |
| 79 | \( 1 + 7.33T + 79T^{2} \) |
| 83 | \( 1 + 8.32T + 83T^{2} \) |
| 89 | \( 1 - 13.5T + 89T^{2} \) |
| 97 | \( 1 + 7.83T + 97T^{2} \) |
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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
−7.58321243888235596129821135800, −7.12226695456431188278024949875, −6.31114603329833797337362154568, −5.77927732917681737026692716288, −5.27288193137734556227974655274, −4.47625231756729024719877778905, −3.48617716682372907270574098783, −2.99267196603154114743720235527, −1.69405882352552328190478774719, −0.990368982624148812032734833212,
0.990368982624148812032734833212, 1.69405882352552328190478774719, 2.99267196603154114743720235527, 3.48617716682372907270574098783, 4.47625231756729024719877778905, 5.27288193137734556227974655274, 5.77927732917681737026692716288, 6.31114603329833797337362154568, 7.12226695456431188278024949875, 7.58321243888235596129821135800
