| L(s) = 1 | + 2.27·3-s − 1.62·5-s − 3.69·7-s + 2.19·9-s − 5.39·11-s − 0.869·13-s − 3.69·15-s + 3.44·17-s + 5.99·19-s − 8.43·21-s − 2.36·25-s − 1.82·27-s + 6.59·29-s + 6.66·31-s − 12.2·33-s + 6.00·35-s − 1.89·37-s − 1.98·39-s + 8.09·41-s + 12.3·43-s − 3.56·45-s − 3.55·47-s + 6.68·49-s + 7.85·51-s − 8.82·53-s + 8.75·55-s + 13.6·57-s + ⋯ |
| L(s) = 1 | + 1.31·3-s − 0.725·5-s − 1.39·7-s + 0.732·9-s − 1.62·11-s − 0.241·13-s − 0.955·15-s + 0.835·17-s + 1.37·19-s − 1.84·21-s − 0.473·25-s − 0.351·27-s + 1.22·29-s + 1.19·31-s − 2.14·33-s + 1.01·35-s − 0.311·37-s − 0.317·39-s + 1.26·41-s + 1.88·43-s − 0.531·45-s − 0.518·47-s + 0.955·49-s + 1.09·51-s − 1.21·53-s + 1.17·55-s + 1.80·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.818143481\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.818143481\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 2.27T + 3T^{2} \) |
| 5 | \( 1 + 1.62T + 5T^{2} \) |
| 7 | \( 1 + 3.69T + 7T^{2} \) |
| 11 | \( 1 + 5.39T + 11T^{2} \) |
| 13 | \( 1 + 0.869T + 13T^{2} \) |
| 17 | \( 1 - 3.44T + 17T^{2} \) |
| 19 | \( 1 - 5.99T + 19T^{2} \) |
| 29 | \( 1 - 6.59T + 29T^{2} \) |
| 31 | \( 1 - 6.66T + 31T^{2} \) |
| 37 | \( 1 + 1.89T + 37T^{2} \) |
| 41 | \( 1 - 8.09T + 41T^{2} \) |
| 43 | \( 1 - 12.3T + 43T^{2} \) |
| 47 | \( 1 + 3.55T + 47T^{2} \) |
| 53 | \( 1 + 8.82T + 53T^{2} \) |
| 59 | \( 1 - 12.5T + 59T^{2} \) |
| 61 | \( 1 + 7.43T + 61T^{2} \) |
| 67 | \( 1 + 7.08T + 67T^{2} \) |
| 71 | \( 1 - 0.0791T + 71T^{2} \) |
| 73 | \( 1 - 3.26T + 73T^{2} \) |
| 79 | \( 1 - 9.31T + 79T^{2} \) |
| 83 | \( 1 - 14.6T + 83T^{2} \) |
| 89 | \( 1 + 2.62T + 89T^{2} \) |
| 97 | \( 1 - 3.58T + 97T^{2} \) |
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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
−8.196866873035724874478512532005, −7.75359540655766198232098679992, −7.33641579981887971317453325285, −6.24902249438072554209683883235, −5.43632939043528215005054862059, −4.44516236038477792468198330052, −3.43678398549030136841801169218, −3.01727334083145676058062497666, −2.43451376683400043615265172356, −0.68395167712277210571966003749,
0.68395167712277210571966003749, 2.43451376683400043615265172356, 3.01727334083145676058062497666, 3.43678398549030136841801169218, 4.44516236038477792468198330052, 5.43632939043528215005054862059, 6.24902249438072554209683883235, 7.33641579981887971317453325285, 7.75359540655766198232098679992, 8.196866873035724874478512532005
