L(s) = 1 | − 3-s + 5-s − 2.56·7-s + 9-s + 6.56·11-s + 13-s − 15-s + 5.68·17-s + 5.12·19-s + 2.56·21-s + 7.68·23-s + 25-s − 27-s − 7.12·29-s + 8·31-s − 6.56·33-s − 2.56·35-s − 3.43·37-s − 39-s + 3.43·41-s + 9.12·43-s + 45-s − 6.24·47-s − 0.438·49-s − 5.68·51-s − 10.8·53-s + 6.56·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.968·7-s + 0.333·9-s + 1.97·11-s + 0.277·13-s − 0.258·15-s + 1.37·17-s + 1.17·19-s + 0.558·21-s + 1.60·23-s + 0.200·25-s − 0.192·27-s − 1.32·29-s + 1.43·31-s − 1.14·33-s − 0.432·35-s − 0.565·37-s − 0.160·39-s + 0.536·41-s + 1.39·43-s + 0.149·45-s − 0.911·47-s − 0.0626·49-s − 0.796·51-s − 1.48·53-s + 0.884·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.187306470\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.187306470\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 + 2.56T + 7T^{2} \) |
| 11 | \( 1 - 6.56T + 11T^{2} \) |
| 17 | \( 1 - 5.68T + 17T^{2} \) |
| 19 | \( 1 - 5.12T + 19T^{2} \) |
| 23 | \( 1 - 7.68T + 23T^{2} \) |
| 29 | \( 1 + 7.12T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 3.43T + 37T^{2} \) |
| 41 | \( 1 - 3.43T + 41T^{2} \) |
| 43 | \( 1 - 9.12T + 43T^{2} \) |
| 47 | \( 1 + 6.24T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 + 2.24T + 59T^{2} \) |
| 61 | \( 1 + 0.561T + 61T^{2} \) |
| 67 | \( 1 - 14.2T + 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 + 8.24T + 73T^{2} \) |
| 79 | \( 1 + 3.68T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 - 10.8T + 89T^{2} \) |
| 97 | \( 1 + 2.31T + 97T^{2} \) |
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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
−7.922059431060278027121976813456, −7.11030600969510771388938435815, −6.58079862079426136927180166029, −5.98265502658912872714400535028, −5.37077405687104632755999337010, −4.43358245539731333966154742743, −3.50321435624880108622769880135, −3.03649822673027567637355638549, −1.48289608670741764807695844685, −0.906156959087911716320027228013,
0.906156959087911716320027228013, 1.48289608670741764807695844685, 3.03649822673027567637355638549, 3.50321435624880108622769880135, 4.43358245539731333966154742743, 5.37077405687104632755999337010, 5.98265502658912872714400535028, 6.58079862079426136927180166029, 7.11030600969510771388938435815, 7.922059431060278027121976813456