| L(s) = 1 | + 3-s − 4.78·7-s + 9-s + 1.95·11-s + 1.07·13-s − 3.80·17-s + 4.25·19-s − 4.78·21-s − 5.86·23-s + 27-s + 10.5·29-s − 4.31·31-s + 1.95·33-s + 5.65·37-s + 1.07·39-s + 0.858·41-s − 10.8·43-s + 3.00·47-s + 15.8·49-s − 3.80·51-s + 4.22·53-s + 4.25·57-s − 4.77·59-s + 3.63·61-s − 4.78·63-s + 6.93·67-s − 5.86·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.80·7-s + 0.333·9-s + 0.589·11-s + 0.299·13-s − 0.923·17-s + 0.975·19-s − 1.04·21-s − 1.22·23-s + 0.192·27-s + 1.95·29-s − 0.775·31-s + 0.340·33-s + 0.930·37-s + 0.172·39-s + 0.134·41-s − 1.65·43-s + 0.437·47-s + 2.26·49-s − 0.532·51-s + 0.580·53-s + 0.563·57-s − 0.621·59-s + 0.465·61-s − 0.602·63-s + 0.846·67-s − 0.705·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 4.78T + 7T^{2} \) |
| 11 | \( 1 - 1.95T + 11T^{2} \) |
| 13 | \( 1 - 1.07T + 13T^{2} \) |
| 17 | \( 1 + 3.80T + 17T^{2} \) |
| 19 | \( 1 - 4.25T + 19T^{2} \) |
| 23 | \( 1 + 5.86T + 23T^{2} \) |
| 29 | \( 1 - 10.5T + 29T^{2} \) |
| 31 | \( 1 + 4.31T + 31T^{2} \) |
| 37 | \( 1 - 5.65T + 37T^{2} \) |
| 41 | \( 1 - 0.858T + 41T^{2} \) |
| 43 | \( 1 + 10.8T + 43T^{2} \) |
| 47 | \( 1 - 3.00T + 47T^{2} \) |
| 53 | \( 1 - 4.22T + 53T^{2} \) |
| 59 | \( 1 + 4.77T + 59T^{2} \) |
| 61 | \( 1 - 3.63T + 61T^{2} \) |
| 67 | \( 1 - 6.93T + 67T^{2} \) |
| 71 | \( 1 + 11.9T + 71T^{2} \) |
| 73 | \( 1 + 16.9T + 73T^{2} \) |
| 79 | \( 1 + 6.62T + 79T^{2} \) |
| 83 | \( 1 + 2.58T + 83T^{2} \) |
| 89 | \( 1 + 0.832T + 89T^{2} \) |
| 97 | \( 1 + 9.11T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.44910355342888523858224452645, −6.75848554640953909737660280330, −6.33620974367645873328866384875, −5.62108297755608466676270751184, −4.44813906607628822212205590784, −3.83230806007196359795981890729, −3.10508360442687710480401431142, −2.50091655989556031137545569332, −1.27547642180696603633629781565, 0,
1.27547642180696603633629781565, 2.50091655989556031137545569332, 3.10508360442687710480401431142, 3.83230806007196359795981890729, 4.44813906607628822212205590784, 5.62108297755608466676270751184, 6.33620974367645873328866384875, 6.75848554640953909737660280330, 7.44910355342888523858224452645
