| L(s) = 1 | + 2.22e3·3-s − 4.97e5·7-s + 3.35e6·9-s − 5.23e6·11-s − 2.26e7·13-s − 1.13e8·17-s − 8.56e7·19-s − 1.10e9·21-s + 9.79e8·23-s + 3.92e9·27-s − 5.34e8·29-s + 3.41e9·31-s − 1.16e10·33-s + 3.08e10·37-s − 5.03e10·39-s − 7.78e9·41-s + 3.72e10·43-s + 2.23e10·47-s + 1.50e11·49-s − 2.52e11·51-s − 9.47e10·53-s − 1.90e11·57-s − 1.49e11·59-s + 4.53e11·61-s − 1.66e12·63-s + 6.15e11·67-s + 2.17e12·69-s + ⋯ |
| L(s) = 1 | + 1.76·3-s − 1.59·7-s + 2.10·9-s − 0.890·11-s − 1.30·13-s − 1.14·17-s − 0.417·19-s − 2.81·21-s + 1.37·23-s + 1.94·27-s − 0.166·29-s + 0.691·31-s − 1.56·33-s + 1.97·37-s − 2.29·39-s − 0.255·41-s + 0.897·43-s + 0.302·47-s + 1.55·49-s − 2.00·51-s − 0.587·53-s − 0.736·57-s − 0.462·59-s + 1.12·61-s − 3.36·63-s + 0.830·67-s + 2.43·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(2.865130248\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.865130248\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 2.22e3T + 1.59e6T^{2} \) |
| 7 | \( 1 + 4.97e5T + 9.68e10T^{2} \) |
| 11 | \( 1 + 5.23e6T + 3.45e13T^{2} \) |
| 13 | \( 1 + 2.26e7T + 3.02e14T^{2} \) |
| 17 | \( 1 + 1.13e8T + 9.90e15T^{2} \) |
| 19 | \( 1 + 8.56e7T + 4.20e16T^{2} \) |
| 23 | \( 1 - 9.79e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + 5.34e8T + 1.02e19T^{2} \) |
| 31 | \( 1 - 3.41e9T + 2.44e19T^{2} \) |
| 37 | \( 1 - 3.08e10T + 2.43e20T^{2} \) |
| 41 | \( 1 + 7.78e9T + 9.25e20T^{2} \) |
| 43 | \( 1 - 3.72e10T + 1.71e21T^{2} \) |
| 47 | \( 1 - 2.23e10T + 5.46e21T^{2} \) |
| 53 | \( 1 + 9.47e10T + 2.60e22T^{2} \) |
| 59 | \( 1 + 1.49e11T + 1.04e23T^{2} \) |
| 61 | \( 1 - 4.53e11T + 1.61e23T^{2} \) |
| 67 | \( 1 - 6.15e11T + 5.48e23T^{2} \) |
| 71 | \( 1 - 1.30e12T + 1.16e24T^{2} \) |
| 73 | \( 1 - 2.03e12T + 1.67e24T^{2} \) |
| 79 | \( 1 - 2.09e12T + 4.66e24T^{2} \) |
| 83 | \( 1 + 1.10e11T + 8.87e24T^{2} \) |
| 89 | \( 1 - 6.25e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + 5.81e12T + 6.73e25T^{2} \) |
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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
−9.645106059096092793385870911914, −9.376421322045414292776423006911, −8.254106670939481061257894470518, −7.31966899833945001455442737738, −6.50986684616669302042649312984, −4.80817915235490702114718442817, −3.70516462911431464323883106495, −2.60271016558388576271154000467, −2.46151389717215521112816790226, −0.59889833358404253340015374823,
0.59889833358404253340015374823, 2.46151389717215521112816790226, 2.60271016558388576271154000467, 3.70516462911431464323883106495, 4.80817915235490702114718442817, 6.50986684616669302042649312984, 7.31966899833945001455442737738, 8.254106670939481061257894470518, 9.376421322045414292776423006911, 9.645106059096092793385870911914
