| L(s) = 1 | + 3-s + 4.26·5-s + 0.932·7-s + 9-s + 11-s + 13-s + 4.26·15-s + 7.19·17-s + 6.90·19-s + 0.932·21-s − 2.93·23-s + 13.1·25-s + 27-s − 9.88·29-s − 8.23·31-s + 33-s + 3.97·35-s − 1.86·37-s + 39-s + 1.06·41-s − 5.30·43-s + 4.26·45-s − 11.1·47-s − 6.13·49-s + 7.19·51-s + 9.24·53-s + 4.26·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.90·5-s + 0.352·7-s + 0.333·9-s + 0.301·11-s + 0.277·13-s + 1.10·15-s + 1.74·17-s + 1.58·19-s + 0.203·21-s − 0.611·23-s + 2.63·25-s + 0.192·27-s − 1.83·29-s − 1.47·31-s + 0.174·33-s + 0.671·35-s − 0.306·37-s + 0.160·39-s + 0.166·41-s − 0.809·43-s + 0.635·45-s − 1.61·47-s − 0.875·49-s + 1.00·51-s + 1.26·53-s + 0.574·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.945936773\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.945936773\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 4.26T + 5T^{2} \) |
| 7 | \( 1 - 0.932T + 7T^{2} \) |
| 17 | \( 1 - 7.19T + 17T^{2} \) |
| 19 | \( 1 - 6.90T + 19T^{2} \) |
| 23 | \( 1 + 2.93T + 23T^{2} \) |
| 29 | \( 1 + 9.88T + 29T^{2} \) |
| 31 | \( 1 + 8.23T + 31T^{2} \) |
| 37 | \( 1 + 1.86T + 37T^{2} \) |
| 41 | \( 1 - 1.06T + 41T^{2} \) |
| 43 | \( 1 + 5.30T + 43T^{2} \) |
| 47 | \( 1 + 11.1T + 47T^{2} \) |
| 53 | \( 1 - 9.24T + 53T^{2} \) |
| 59 | \( 1 - 9.26T + 59T^{2} \) |
| 61 | \( 1 + 14.5T + 61T^{2} \) |
| 67 | \( 1 + 2.26T + 67T^{2} \) |
| 71 | \( 1 + 7.23T + 71T^{2} \) |
| 73 | \( 1 + 6.17T + 73T^{2} \) |
| 79 | \( 1 + 2.62T + 79T^{2} \) |
| 83 | \( 1 + 8.52T + 83T^{2} \) |
| 89 | \( 1 - 14.8T + 89T^{2} \) |
| 97 | \( 1 - 6.71T + 97T^{2} \) |
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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
−8.827617146207419840637240206467, −7.78041068619259400479771583532, −7.23871734455867759417152451078, −6.19341823460748886518341447107, −5.51991744798561283260441994854, −5.12296594422693418466372212350, −3.66985601006490282570870239415, −3.01009189152801111287279484734, −1.79606729833594650756065354432, −1.40069348735935799722051889392,
1.40069348735935799722051889392, 1.79606729833594650756065354432, 3.01009189152801111287279484734, 3.66985601006490282570870239415, 5.12296594422693418466372212350, 5.51991744798561283260441994854, 6.19341823460748886518341447107, 7.23871734455867759417152451078, 7.78041068619259400479771583532, 8.827617146207419840637240206467
