| L(s) = 1 | + 3.04·3-s + 6.25·9-s + 3.20·11-s + 6.25·13-s − 1.78·17-s + 3.67·19-s + 3.21·23-s + 9.92·27-s + 2.88·29-s − 2.81·31-s + 9.73·33-s + 5.08·37-s + 19.0·39-s + 0.899·41-s − 7.26·43-s + 2.52·47-s − 5.42·51-s − 9.04·53-s + 11.1·57-s + 5.58·59-s − 10.9·61-s + 1.40·67-s + 9.78·69-s − 15.3·71-s + 7.01·73-s + 0.186·79-s + 11.4·81-s + ⋯ |
| L(s) = 1 | + 1.75·3-s + 2.08·9-s + 0.965·11-s + 1.73·13-s − 0.432·17-s + 0.843·19-s + 0.670·23-s + 1.90·27-s + 0.535·29-s − 0.506·31-s + 1.69·33-s + 0.835·37-s + 3.05·39-s + 0.140·41-s − 1.10·43-s + 0.367·47-s − 0.759·51-s − 1.24·53-s + 1.48·57-s + 0.727·59-s − 1.39·61-s + 0.171·67-s + 1.17·69-s − 1.82·71-s + 0.821·73-s + 0.0210·79-s + 1.26·81-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
| \(L(1)\) |
\(\approx\) |
\(5.457072774\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.457072774\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.04T + 3T^{2} \) |
| 11 | \( 1 - 3.20T + 11T^{2} \) |
| 13 | \( 1 - 6.25T + 13T^{2} \) |
| 17 | \( 1 + 1.78T + 17T^{2} \) |
| 19 | \( 1 - 3.67T + 19T^{2} \) |
| 23 | \( 1 - 3.21T + 23T^{2} \) |
| 29 | \( 1 - 2.88T + 29T^{2} \) |
| 31 | \( 1 + 2.81T + 31T^{2} \) |
| 37 | \( 1 - 5.08T + 37T^{2} \) |
| 41 | \( 1 - 0.899T + 41T^{2} \) |
| 43 | \( 1 + 7.26T + 43T^{2} \) |
| 47 | \( 1 - 2.52T + 47T^{2} \) |
| 53 | \( 1 + 9.04T + 53T^{2} \) |
| 59 | \( 1 - 5.58T + 59T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 - 1.40T + 67T^{2} \) |
| 71 | \( 1 + 15.3T + 71T^{2} \) |
| 73 | \( 1 - 7.01T + 73T^{2} \) |
| 79 | \( 1 - 0.186T + 79T^{2} \) |
| 83 | \( 1 - 0.134T + 83T^{2} \) |
| 89 | \( 1 + 18.4T + 89T^{2} \) |
| 97 | \( 1 + 9.75T + 97T^{2} \) |
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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
−7.86367996250105103194877047466, −7.06864442143503007403719197123, −6.53430454098694809072069731758, −5.71956582004350012383070096086, −4.57947349722651625827044112015, −3.97632187799480587580527527822, −3.33383910431295756625557369296, −2.81885356316534144163888389832, −1.68083531314568127030566353276, −1.17053015083395442087841755255,
1.17053015083395442087841755255, 1.68083531314568127030566353276, 2.81885356316534144163888389832, 3.33383910431295756625557369296, 3.97632187799480587580527527822, 4.57947349722651625827044112015, 5.71956582004350012383070096086, 6.53430454098694809072069731758, 7.06864442143503007403719197123, 7.86367996250105103194877047466
