L(s) = 1 | − 2-s + 4-s + 5-s + 7-s − 8-s − 10-s + 5·11-s − 14-s + 16-s − 17-s + 19-s + 20-s − 5·22-s + 25-s + 28-s + 2·29-s − 32-s + 34-s + 35-s + 7·37-s − 38-s − 40-s − 2·41-s − 8·43-s + 5·44-s − 3·47-s − 6·49-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 + 1.50·11-s − 0.267·14-s + 1/4·16-s − 0.242·17-s + 0.229·19-s + 0.223·20-s − 1.06·22-s + 1/5·25-s + 0.188·28-s + 0.371·29-s − 0.176·32-s + 0.171·34-s + 0.169·35-s + 1.15·37-s − 0.162·38-s − 0.158·40-s − 0.312·41-s − 1.21·43-s + 0.753·44-s − 0.437·47-s − 6/7·49-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 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−12.97754860996997, −12.58079277858257, −11.86118249867548, −11.67839742074456, −11.23121269873561, −10.74484417643553, −10.17955829181736, −9.703017902587891, −9.374543726037015, −8.950758506580617, −8.424437310010084, −7.991325178677069, −7.503822310097419, −6.823655894306705, −6.454782877006635, −6.195972605012203, −5.465336619757120, −4.845004669332230, −4.450398716659199, −3.680641216084485, −3.264626528802132, −2.543488674463619, −1.894126071131401, −1.416684964542443, −0.9308243946217461, 0,
0.9308243946217461, 1.416684964542443, 1.894126071131401, 2.543488674463619, 3.264626528802132, 3.680641216084485, 4.450398716659199, 4.845004669332230, 5.465336619757120, 6.195972605012203, 6.454782877006635, 6.823655894306705, 7.503822310097419, 7.991325178677069, 8.424437310010084, 8.950758506580617, 9.374543726037015, 9.703017902587891, 10.17955829181736, 10.74484417643553, 11.23121269873561, 11.67839742074456, 11.86118249867548, 12.58079277858257, 12.97754860996997