L(s) = 1 | − 2-s + 4-s − 8-s − 5·11-s − 13-s + 16-s − 7·17-s − 19-s + 5·22-s + 6·23-s + 26-s + 6·29-s − 31-s − 32-s + 7·34-s + 38-s − 2·41-s − 2·43-s − 5·44-s − 6·46-s − 52-s + 6·53-s − 6·58-s + 4·59-s − 3·61-s + 62-s + 64-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 1.50·11-s − 0.277·13-s + 1/4·16-s − 1.69·17-s − 0.229·19-s + 1.06·22-s + 1.25·23-s + 0.196·26-s + 1.11·29-s − 0.179·31-s − 0.176·32-s + 1.20·34-s + 0.162·38-s − 0.312·41-s − 0.304·43-s − 0.753·44-s − 0.884·46-s − 0.138·52-s + 0.824·53-s − 0.787·58-s + 0.520·59-s − 0.384·61-s + 0.127·62-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7482472826\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7482472826\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−12.79804419964039, −12.23164487473347, −11.67306988885360, −11.29277091673247, −10.67379492640153, −10.51861110936248, −10.10220203408590, −9.427032208360821, −8.924359636519533, −8.681417367883293, −8.092740141475812, −7.700926575354183, −7.079463181560452, −6.792275116496762, −6.267503187370605, −5.610645982557041, −5.096261529649986, −4.653697477461717, −4.165568692327118, −3.232637340997199, −2.844254706516950, −2.313254571830753, −1.858074290667846, −0.9620095850579657, −0.2939673211738746,
0.2939673211738746, 0.9620095850579657, 1.858074290667846, 2.313254571830753, 2.844254706516950, 3.232637340997199, 4.165568692327118, 4.653697477461717, 5.096261529649986, 5.610645982557041, 6.267503187370605, 6.792275116496762, 7.079463181560452, 7.700926575354183, 8.092740141475812, 8.681417367883293, 8.924359636519533, 9.427032208360821, 10.10220203408590, 10.51861110936248, 10.67379492640153, 11.29277091673247, 11.67306988885360, 12.23164487473347, 12.79804419964039