| L(s) = 1 | + 3.18·3-s + 0.0919·5-s + 3.89·7-s + 7.13·9-s − 2.60·13-s + 0.292·15-s + 0.530·17-s + 5.03·19-s + 12.3·21-s − 3.05·23-s − 4.99·25-s + 13.1·27-s + 6.60·29-s − 4.76·31-s + 0.357·35-s − 8.24·37-s − 8.29·39-s + 2.90·41-s + 6.01·43-s + 0.655·45-s + 4.36·47-s + 8.15·49-s + 1.68·51-s − 10.8·53-s + 16.0·57-s + 11.2·59-s + 11.4·61-s + ⋯ |
| L(s) = 1 | + 1.83·3-s + 0.0410·5-s + 1.47·7-s + 2.37·9-s − 0.722·13-s + 0.0755·15-s + 0.128·17-s + 1.15·19-s + 2.70·21-s − 0.636·23-s − 0.998·25-s + 2.53·27-s + 1.22·29-s − 0.855·31-s + 0.0604·35-s − 1.35·37-s − 1.32·39-s + 0.453·41-s + 0.917·43-s + 0.0977·45-s + 0.636·47-s + 1.16·49-s + 0.236·51-s − 1.49·53-s + 2.12·57-s + 1.45·59-s + 1.46·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.434863315\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.434863315\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 3.18T + 3T^{2} \) |
| 5 | \( 1 - 0.0919T + 5T^{2} \) |
| 7 | \( 1 - 3.89T + 7T^{2} \) |
| 13 | \( 1 + 2.60T + 13T^{2} \) |
| 17 | \( 1 - 0.530T + 17T^{2} \) |
| 19 | \( 1 - 5.03T + 19T^{2} \) |
| 23 | \( 1 + 3.05T + 23T^{2} \) |
| 29 | \( 1 - 6.60T + 29T^{2} \) |
| 31 | \( 1 + 4.76T + 31T^{2} \) |
| 37 | \( 1 + 8.24T + 37T^{2} \) |
| 41 | \( 1 - 2.90T + 41T^{2} \) |
| 43 | \( 1 - 6.01T + 43T^{2} \) |
| 47 | \( 1 - 4.36T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 - 11.2T + 59T^{2} \) |
| 61 | \( 1 - 11.4T + 61T^{2} \) |
| 67 | \( 1 - 7.33T + 67T^{2} \) |
| 71 | \( 1 - 11.2T + 71T^{2} \) |
| 73 | \( 1 - 2.52T + 73T^{2} \) |
| 79 | \( 1 + 8.63T + 79T^{2} \) |
| 83 | \( 1 - 1.53T + 83T^{2} \) |
| 89 | \( 1 + 3.93T + 89T^{2} \) |
| 97 | \( 1 + 4.80T + 97T^{2} \) |
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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
−8.070581780565175928244671826018, −7.41232134287471945031979818148, −6.89758613278486497002072502590, −5.55855140753764037865066603936, −4.91990866738990760732684886247, −4.11925468702783211917650256263, −3.52283840962537531178972256478, −2.52437475353730269026805866756, −2.01636540565932902298352651784, −1.16530348880852832515136139027,
1.16530348880852832515136139027, 2.01636540565932902298352651784, 2.52437475353730269026805866756, 3.52283840962537531178972256478, 4.11925468702783211917650256263, 4.91990866738990760732684886247, 5.55855140753764037865066603936, 6.89758613278486497002072502590, 7.41232134287471945031979818148, 8.070581780565175928244671826018
