| L(s) = 1 | − 2·7-s + 11-s − 4·13-s − 2·17-s + 6·23-s − 10·29-s + 8·31-s + 2·37-s − 2·41-s + 4·43-s + 2·47-s − 3·49-s + 4·53-s − 8·61-s − 12·67-s + 2·71-s + 6·73-s − 2·77-s − 10·79-s − 4·83-s − 10·89-s + 8·91-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 0.301·11-s − 1.10·13-s − 0.485·17-s + 1.25·23-s − 1.85·29-s + 1.43·31-s + 0.328·37-s − 0.312·41-s + 0.609·43-s + 0.291·47-s − 3/7·49-s + 0.549·53-s − 1.02·61-s − 1.46·67-s + 0.237·71-s + 0.702·73-s − 0.227·77-s − 1.12·79-s − 0.439·83-s − 1.05·89-s + 0.838·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.230694489\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.230694489\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.90172009612533, −14.29713393185963, −13.72017295598000, −13.15418632295282, −12.76817133474780, −12.29063488875269, −11.59545480018904, −11.24932455425058, −10.51870496970071, −10.01230007324743, −9.452101224323053, −9.089209127682775, −8.501208011042061, −7.645581752707029, −7.261693586177805, −6.722125441157602, −6.101022370586261, −5.532600364174996, −4.766972464385708, −4.330369819031685, −3.493251665291307, −2.894342965093724, −2.304859494785706, −1.399099992168693, −0.4079152530575440,
0.4079152530575440, 1.399099992168693, 2.304859494785706, 2.894342965093724, 3.493251665291307, 4.330369819031685, 4.766972464385708, 5.532600364174996, 6.101022370586261, 6.722125441157602, 7.261693586177805, 7.645581752707029, 8.501208011042061, 9.089209127682775, 9.452101224323053, 10.01230007324743, 10.51870496970071, 11.24932455425058, 11.59545480018904, 12.29063488875269, 12.76817133474780, 13.15418632295282, 13.72017295598000, 14.29713393185963, 14.90172009612533
