| L(s) = 1 | − 0.295·3-s + 4.07·5-s − 1.80·7-s − 2.91·9-s + 3.73·13-s − 1.20·15-s − 4.31·17-s + 19-s + 0.531·21-s − 4.93·23-s + 11.6·25-s + 1.74·27-s + 8.16·29-s − 8.81·31-s − 7.34·35-s − 8.92·37-s − 1.10·39-s − 7.37·41-s + 5.87·43-s − 11.8·45-s − 9.19·47-s − 3.75·49-s + 1.27·51-s + 7.76·53-s − 0.295·57-s − 6.16·59-s + 4.41·61-s + ⋯ |
| L(s) = 1 | − 0.170·3-s + 1.82·5-s − 0.680·7-s − 0.970·9-s + 1.03·13-s − 0.311·15-s − 1.04·17-s + 0.229·19-s + 0.116·21-s − 1.02·23-s + 2.32·25-s + 0.336·27-s + 1.51·29-s − 1.58·31-s − 1.24·35-s − 1.46·37-s − 0.176·39-s − 1.15·41-s + 0.895·43-s − 1.77·45-s − 1.34·47-s − 0.536·49-s + 0.178·51-s + 1.06·53-s − 0.0391·57-s − 0.802·59-s + 0.565·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 0.295T + 3T^{2} \) |
| 5 | \( 1 - 4.07T + 5T^{2} \) |
| 7 | \( 1 + 1.80T + 7T^{2} \) |
| 13 | \( 1 - 3.73T + 13T^{2} \) |
| 17 | \( 1 + 4.31T + 17T^{2} \) |
| 23 | \( 1 + 4.93T + 23T^{2} \) |
| 29 | \( 1 - 8.16T + 29T^{2} \) |
| 31 | \( 1 + 8.81T + 31T^{2} \) |
| 37 | \( 1 + 8.92T + 37T^{2} \) |
| 41 | \( 1 + 7.37T + 41T^{2} \) |
| 43 | \( 1 - 5.87T + 43T^{2} \) |
| 47 | \( 1 + 9.19T + 47T^{2} \) |
| 53 | \( 1 - 7.76T + 53T^{2} \) |
| 59 | \( 1 + 6.16T + 59T^{2} \) |
| 61 | \( 1 - 4.41T + 61T^{2} \) |
| 67 | \( 1 - 0.555T + 67T^{2} \) |
| 71 | \( 1 - 1.15T + 71T^{2} \) |
| 73 | \( 1 - 0.509T + 73T^{2} \) |
| 79 | \( 1 + 8.03T + 79T^{2} \) |
| 83 | \( 1 + 6.29T + 83T^{2} \) |
| 89 | \( 1 + 6.85T + 89T^{2} \) |
| 97 | \( 1 + 2.06T + 97T^{2} \) |
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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
−7.04548174716235541107820961139, −6.51232004696830219312866400341, −6.02170609674803808460675237289, −5.52583004616327306018644208640, −4.83144171154273973758219358900, −3.68354291154793177970639846891, −2.93651609799288764007663444141, −2.15932241430727932970281575733, −1.40875957510422810907271841970, 0,
1.40875957510422810907271841970, 2.15932241430727932970281575733, 2.93651609799288764007663444141, 3.68354291154793177970639846891, 4.83144171154273973758219358900, 5.52583004616327306018644208640, 6.02170609674803808460675237289, 6.51232004696830219312866400341, 7.04548174716235541107820961139
