L(s) = 1 | − 2.34·3-s + 1.19·7-s + 2.48·9-s − 4.97·11-s + 6.63·13-s − 1.48·17-s − 19-s − 2.80·21-s − 0.510·23-s + 1.19·27-s − 7.88·29-s + 2.97·31-s + 11.6·33-s + 7.14·37-s − 15.5·39-s + 1.66·41-s − 6.39·43-s − 9.95·47-s − 5.56·49-s + 3.48·51-s + 11.4·53-s + 2.34·57-s + 11.8·59-s + 3.66·61-s + 2.97·63-s + 7.61·67-s + 1.19·69-s + ⋯ |
L(s) = 1 | − 1.35·3-s + 0.452·7-s + 0.829·9-s − 1.50·11-s + 1.84·13-s − 0.361·17-s − 0.229·19-s − 0.611·21-s − 0.106·23-s + 0.230·27-s − 1.46·29-s + 0.534·31-s + 2.03·33-s + 1.17·37-s − 2.48·39-s + 0.259·41-s − 0.974·43-s − 1.45·47-s − 0.795·49-s + 0.488·51-s + 1.56·53-s + 0.310·57-s + 1.54·59-s + 0.469·61-s + 0.375·63-s + 0.930·67-s + 0.144·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 + 2.34T + 3T^{2} \) |
| 7 | \( 1 - 1.19T + 7T^{2} \) |
| 11 | \( 1 + 4.97T + 11T^{2} \) |
| 13 | \( 1 - 6.63T + 13T^{2} \) |
| 17 | \( 1 + 1.48T + 17T^{2} \) |
| 23 | \( 1 + 0.510T + 23T^{2} \) |
| 29 | \( 1 + 7.88T + 29T^{2} \) |
| 31 | \( 1 - 2.97T + 31T^{2} \) |
| 37 | \( 1 - 7.14T + 37T^{2} \) |
| 41 | \( 1 - 1.66T + 41T^{2} \) |
| 43 | \( 1 + 6.39T + 43T^{2} \) |
| 47 | \( 1 + 9.95T + 47T^{2} \) |
| 53 | \( 1 - 11.4T + 53T^{2} \) |
| 59 | \( 1 - 11.8T + 59T^{2} \) |
| 61 | \( 1 - 3.66T + 61T^{2} \) |
| 67 | \( 1 - 7.61T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 13.8T + 73T^{2} \) |
| 79 | \( 1 + 12.6T + 79T^{2} \) |
| 83 | \( 1 + 8.68T + 83T^{2} \) |
| 89 | \( 1 + 4.87T + 89T^{2} \) |
| 97 | \( 1 - 6.81T + 97T^{2} \) |
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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
−7.42877596048540592806071141937, −6.70151378569186683147444614844, −5.90514988057904074176330268870, −5.61467721026013841803444097749, −4.84953238165544099428324539667, −4.14191629345835874962304382767, −3.18368769000238892236909614672, −2.08220868731188979112641009485, −1.06023726517714470950466486405, 0,
1.06023726517714470950466486405, 2.08220868731188979112641009485, 3.18368769000238892236909614672, 4.14191629345835874962304382767, 4.84953238165544099428324539667, 5.61467721026013841803444097749, 5.90514988057904074176330268870, 6.70151378569186683147444614844, 7.42877596048540592806071141937