| L(s) = 1 | − 2-s + 4-s − 5-s − 7-s − 8-s + 10-s + 14-s + 16-s + 17-s − 20-s + 25-s − 28-s + 6·29-s + 8·31-s − 32-s − 34-s + 35-s − 4·37-s + 40-s + 10·41-s + 8·43-s − 6·49-s − 50-s + 4·53-s + 56-s − 6·58-s + 9·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s − 0.353·8-s + 0.316·10-s + 0.267·14-s + 1/4·16-s + 0.242·17-s − 0.223·20-s + 1/5·25-s − 0.188·28-s + 1.11·29-s + 1.43·31-s − 0.176·32-s − 0.171·34-s + 0.169·35-s − 0.657·37-s + 0.158·40-s + 1.56·41-s + 1.21·43-s − 6/7·49-s − 0.141·50-s + 0.549·53-s + 0.133·56-s − 0.787·58-s + 1.17·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 258570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 258570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−12.96755515204246, −12.45771039766817, −12.10017817937110, −11.59961522406113, −11.25599094190283, −10.59668469372947, −10.26738649573652, −9.877379266876963, −9.283713459215638, −8.888343294294534, −8.385597075806336, −7.917058455430727, −7.559232804224937, −6.936892881627814, −6.511469728109861, −6.106264785072954, −5.461081010018680, −4.890769194421085, −4.299067779767340, −3.764472814883847, −3.189337352493291, −2.548740676841288, −2.229968336673835, −1.102390042212853, −0.8825863404514489, 0,
0.8825863404514489, 1.102390042212853, 2.229968336673835, 2.548740676841288, 3.189337352493291, 3.764472814883847, 4.299067779767340, 4.890769194421085, 5.461081010018680, 6.106264785072954, 6.511469728109861, 6.936892881627814, 7.559232804224937, 7.917058455430727, 8.385597075806336, 8.888343294294534, 9.283713459215638, 9.877379266876963, 10.26738649573652, 10.59668469372947, 11.25599094190283, 11.59961522406113, 12.10017817937110, 12.45771039766817, 12.96755515204246
