| L(s) = 1 | + 2.43·3-s + 3.60·7-s + 2.90·9-s − 2.37·11-s − 3.96·13-s + 4.28·17-s + 19-s + 8.76·21-s − 1.07·23-s − 0.227·27-s + 9.47·29-s + 6.38·31-s − 5.78·33-s + 2.04·37-s − 9.63·39-s − 4.38·41-s + 7.86·43-s + 3.83·47-s + 6.01·49-s + 10.4·51-s − 11.7·53-s + 2.43·57-s + 4.59·59-s + 6.62·61-s + 10.4·63-s + 7.02·67-s − 2.60·69-s + ⋯ |
| L(s) = 1 | + 1.40·3-s + 1.36·7-s + 0.968·9-s − 0.717·11-s − 1.10·13-s + 1.03·17-s + 0.229·19-s + 1.91·21-s − 0.223·23-s − 0.0437·27-s + 1.75·29-s + 1.14·31-s − 1.00·33-s + 0.336·37-s − 1.54·39-s − 0.684·41-s + 1.19·43-s + 0.559·47-s + 0.859·49-s + 1.45·51-s − 1.61·53-s + 0.321·57-s + 0.598·59-s + 0.848·61-s + 1.32·63-s + 0.858·67-s − 0.314·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.690815542\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.690815542\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 2.43T + 3T^{2} \) |
| 7 | \( 1 - 3.60T + 7T^{2} \) |
| 11 | \( 1 + 2.37T + 11T^{2} \) |
| 13 | \( 1 + 3.96T + 13T^{2} \) |
| 17 | \( 1 - 4.28T + 17T^{2} \) |
| 23 | \( 1 + 1.07T + 23T^{2} \) |
| 29 | \( 1 - 9.47T + 29T^{2} \) |
| 31 | \( 1 - 6.38T + 31T^{2} \) |
| 37 | \( 1 - 2.04T + 37T^{2} \) |
| 41 | \( 1 + 4.38T + 41T^{2} \) |
| 43 | \( 1 - 7.86T + 43T^{2} \) |
| 47 | \( 1 - 3.83T + 47T^{2} \) |
| 53 | \( 1 + 11.7T + 53T^{2} \) |
| 59 | \( 1 - 4.59T + 59T^{2} \) |
| 61 | \( 1 - 6.62T + 61T^{2} \) |
| 67 | \( 1 - 7.02T + 67T^{2} \) |
| 71 | \( 1 - 4.99T + 71T^{2} \) |
| 73 | \( 1 + 2.93T + 73T^{2} \) |
| 79 | \( 1 - 0.860T + 79T^{2} \) |
| 83 | \( 1 - 11.9T + 83T^{2} \) |
| 89 | \( 1 - 13.6T + 89T^{2} \) |
| 97 | \( 1 - 18.5T + 97T^{2} \) |
| 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
−8.316572078009453769391925113215, −7.82950306131341214747112920926, −7.53999669555286333496499168664, −6.38634163047286501165859753042, −5.16982521813512445110662776189, −4.78287478025222315653825921355, −3.76274819037525214049690582969, −2.74317353628477875567174612397, −2.28931107043873751754527541729, −1.10965333754949256394053909324,
1.10965333754949256394053909324, 2.28931107043873751754527541729, 2.74317353628477875567174612397, 3.76274819037525214049690582969, 4.78287478025222315653825921355, 5.16982521813512445110662776189, 6.38634163047286501165859753042, 7.53999669555286333496499168664, 7.82950306131341214747112920926, 8.316572078009453769391925113215
