L(s) = 1 | − 3-s + 5-s + 7-s − 2·9-s − 13-s − 15-s − 3·17-s − 19-s − 21-s − 3·23-s + 25-s + 5·27-s − 3·29-s − 2·31-s + 35-s − 10·37-s + 39-s + 6·41-s − 2·43-s − 2·45-s − 6·49-s + 3·51-s + 3·53-s + 57-s − 3·59-s + 8·61-s − 2·63-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s − 2/3·9-s − 0.277·13-s − 0.258·15-s − 0.727·17-s − 0.229·19-s − 0.218·21-s − 0.625·23-s + 1/5·25-s + 0.962·27-s − 0.557·29-s − 0.359·31-s + 0.169·35-s − 1.64·37-s + 0.160·39-s + 0.937·41-s − 0.304·43-s − 0.298·45-s − 6/7·49-s + 0.420·51-s + 0.412·53-s + 0.132·57-s − 0.390·59-s + 1.02·61-s − 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−8.985992202019533467038785145213, −8.412887815220587989400052076943, −7.36939617775242477603507597422, −6.52061944989712916278916517958, −5.72097876958001523126129137906, −5.07343753056513176169564111607, −4.07122685411677794871025218409, −2.78657644407780172553397041532, −1.71285992945721434713256047961, 0,
1.71285992945721434713256047961, 2.78657644407780172553397041532, 4.07122685411677794871025218409, 5.07343753056513176169564111607, 5.72097876958001523126129137906, 6.52061944989712916278916517958, 7.36939617775242477603507597422, 8.412887815220587989400052076943, 8.985992202019533467038785145213