L(s) = 1 | + 2-s + 4-s − 2·5-s + 7-s + 8-s − 2·10-s − 4.82·11-s + 5.65·13-s + 14-s + 16-s − 6.82·17-s − 19-s − 2·20-s − 4.82·22-s − 4·23-s − 25-s + 5.65·26-s + 28-s − 2·29-s − 4.82·31-s + 32-s − 6.82·34-s − 2·35-s − 8.48·37-s − 38-s − 2·40-s + 3.65·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.894·5-s + 0.377·7-s + 0.353·8-s − 0.632·10-s − 1.45·11-s + 1.56·13-s + 0.267·14-s + 0.250·16-s − 1.65·17-s − 0.229·19-s − 0.447·20-s − 1.02·22-s − 0.834·23-s − 0.200·25-s + 1.10·26-s + 0.188·28-s − 0.371·29-s − 0.867·31-s + 0.176·32-s − 1.17·34-s − 0.338·35-s − 1.39·37-s − 0.162·38-s − 0.316·40-s + 0.571·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 + T \) |
good | 5 | \( 1 + 2T + 5T^{2} \) |
| 11 | \( 1 + 4.82T + 11T^{2} \) |
| 13 | \( 1 - 5.65T + 13T^{2} \) |
| 17 | \( 1 + 6.82T + 17T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 4.82T + 31T^{2} \) |
| 37 | \( 1 + 8.48T + 37T^{2} \) |
| 41 | \( 1 - 3.65T + 41T^{2} \) |
| 43 | \( 1 - 9.65T + 43T^{2} \) |
| 47 | \( 1 - 4.48T + 47T^{2} \) |
| 53 | \( 1 - 7.65T + 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 + 13.3T + 61T^{2} \) |
| 67 | \( 1 + 6.82T + 67T^{2} \) |
| 71 | \( 1 - 5.65T + 71T^{2} \) |
| 73 | \( 1 + 7.65T + 73T^{2} \) |
| 79 | \( 1 - 7.65T + 79T^{2} \) |
| 83 | \( 1 + 11.6T + 83T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 + 18.4T + 97T^{2} \) |
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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
−8.436271607752536886521661730061, −7.76344836819151572356159532501, −7.10029060411700939382893004477, −6.06063819866439465166710441605, −5.45095517266265691246580779339, −4.33133146033105419204605526577, −3.94589088025960481820071415042, −2.82478785985257348812006980904, −1.80024228504790852269968571759, 0,
1.80024228504790852269968571759, 2.82478785985257348812006980904, 3.94589088025960481820071415042, 4.33133146033105419204605526577, 5.45095517266265691246580779339, 6.06063819866439465166710441605, 7.10029060411700939382893004477, 7.76344836819151572356159532501, 8.436271607752536886521661730061