| L(s) = 1 | + 2.58·3-s + 3.70·9-s + 4.70·11-s − 2.58·13-s − 1.81·17-s + 5.95·19-s + 7.40·23-s + 1.81·27-s − 6.70·29-s − 1.54·31-s + 12.1·33-s + 7.40·37-s − 6.70·39-s + 1.54·41-s + 4·43-s − 10.6·47-s − 4.70·51-s + 11.4·53-s + 15.4·57-s − 13.2·59-s + 2.85·61-s + 8·67-s + 19.1·69-s + 7.40·71-s + 12.4·73-s − 8.10·79-s − 6.40·81-s + ⋯ |
| L(s) = 1 | + 1.49·3-s + 1.23·9-s + 1.41·11-s − 0.717·13-s − 0.440·17-s + 1.36·19-s + 1.54·23-s + 0.349·27-s − 1.24·29-s − 0.277·31-s + 2.11·33-s + 1.21·37-s − 1.07·39-s + 0.241·41-s + 0.609·43-s − 1.54·47-s − 0.658·51-s + 1.56·53-s + 2.04·57-s − 1.72·59-s + 0.366·61-s + 0.977·67-s + 2.30·69-s + 0.878·71-s + 1.45·73-s − 0.911·79-s − 0.711·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.342099882\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.342099882\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 2.58T + 3T^{2} \) |
| 11 | \( 1 - 4.70T + 11T^{2} \) |
| 13 | \( 1 + 2.58T + 13T^{2} \) |
| 17 | \( 1 + 1.81T + 17T^{2} \) |
| 19 | \( 1 - 5.95T + 19T^{2} \) |
| 23 | \( 1 - 7.40T + 23T^{2} \) |
| 29 | \( 1 + 6.70T + 29T^{2} \) |
| 31 | \( 1 + 1.54T + 31T^{2} \) |
| 37 | \( 1 - 7.40T + 37T^{2} \) |
| 41 | \( 1 - 1.54T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 10.6T + 47T^{2} \) |
| 53 | \( 1 - 11.4T + 53T^{2} \) |
| 59 | \( 1 + 13.2T + 59T^{2} \) |
| 61 | \( 1 - 2.85T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 - 7.40T + 71T^{2} \) |
| 73 | \( 1 - 12.4T + 73T^{2} \) |
| 79 | \( 1 + 8.10T + 79T^{2} \) |
| 83 | \( 1 + 2.85T + 83T^{2} \) |
| 89 | \( 1 - 13.4T + 89T^{2} \) |
| 97 | \( 1 - 8.53T + 97T^{2} \) |
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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
−7.64341007233827606993467477152, −7.18687918777839754393628955507, −6.56692727724728592528041745471, −5.56546580461882023139295778575, −4.76160256600655953992683224572, −3.95015247307215715082994888746, −3.37803160866828962358951579461, −2.68689120733514705272663148948, −1.86610295311175798520246165356, −0.963899144468201309344432732510,
0.963899144468201309344432732510, 1.86610295311175798520246165356, 2.68689120733514705272663148948, 3.37803160866828962358951579461, 3.95015247307215715082994888746, 4.76160256600655953992683224572, 5.56546580461882023139295778575, 6.56692727724728592528041745471, 7.18687918777839754393628955507, 7.64341007233827606993467477152
