| L(s) = 1 | − 1.73·3-s − 3.32·7-s + 0.0262·9-s + 5.77·11-s − 1.10·13-s − 0.893·17-s + 2.42·19-s + 5.77·21-s − 23-s + 5.17·27-s + 4.11·29-s − 9.54·31-s − 10.0·33-s + 7.69·37-s + 1.91·39-s + 0.00418·41-s + 9.97·43-s − 10.0·47-s + 4.03·49-s + 1.55·51-s − 6.25·53-s − 4.22·57-s − 10.7·59-s + 10.5·61-s − 0.0871·63-s − 10.9·67-s + 1.73·69-s + ⋯ |
| L(s) = 1 | − 1.00·3-s − 1.25·7-s + 0.00874·9-s + 1.74·11-s − 0.305·13-s − 0.216·17-s + 0.557·19-s + 1.26·21-s − 0.208·23-s + 0.995·27-s + 0.763·29-s − 1.71·31-s − 1.74·33-s + 1.26·37-s + 0.306·39-s + 0.000653·41-s + 1.52·43-s − 1.45·47-s + 0.576·49-s + 0.217·51-s − 0.858·53-s − 0.559·57-s − 1.40·59-s + 1.35·61-s − 0.0109·63-s − 1.33·67-s + 0.209·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + 1.73T + 3T^{2} \) |
| 7 | \( 1 + 3.32T + 7T^{2} \) |
| 11 | \( 1 - 5.77T + 11T^{2} \) |
| 13 | \( 1 + 1.10T + 13T^{2} \) |
| 17 | \( 1 + 0.893T + 17T^{2} \) |
| 19 | \( 1 - 2.42T + 19T^{2} \) |
| 29 | \( 1 - 4.11T + 29T^{2} \) |
| 31 | \( 1 + 9.54T + 31T^{2} \) |
| 37 | \( 1 - 7.69T + 37T^{2} \) |
| 41 | \( 1 - 0.00418T + 41T^{2} \) |
| 43 | \( 1 - 9.97T + 43T^{2} \) |
| 47 | \( 1 + 10.0T + 47T^{2} \) |
| 53 | \( 1 + 6.25T + 53T^{2} \) |
| 59 | \( 1 + 10.7T + 59T^{2} \) |
| 61 | \( 1 - 10.5T + 61T^{2} \) |
| 67 | \( 1 + 10.9T + 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 + 1.89T + 73T^{2} \) |
| 79 | \( 1 + 0.216T + 79T^{2} \) |
| 83 | \( 1 - 5.38T + 83T^{2} \) |
| 89 | \( 1 - 6.00T + 89T^{2} \) |
| 97 | \( 1 + 2.08T + 97T^{2} \) |
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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
−8.937894466922340810965442729063, −7.66713472317423917962462749605, −6.76554154072771086590426334842, −6.29656463929425610469724748996, −5.70439412052745397127830427648, −4.62848342456664722168654221463, −3.75309088737031746016382898324, −2.83922506912535030969575718616, −1.28527266551601642113748567026, 0,
1.28527266551601642113748567026, 2.83922506912535030969575718616, 3.75309088737031746016382898324, 4.62848342456664722168654221463, 5.70439412052745397127830427648, 6.29656463929425610469724748996, 6.76554154072771086590426334842, 7.66713472317423917962462749605, 8.937894466922340810965442729063
