| L(s) = 1 | + 3-s + 3.18·7-s + 9-s + 0.0550·11-s + 5.87·13-s + 5.79·17-s + 5.64·19-s + 3.18·21-s − 4.12·23-s + 27-s + 4.59·29-s − 31-s + 0.0550·33-s + 4.40·37-s + 5.87·39-s − 5.06·41-s + 10.0·43-s − 5.56·47-s + 3.16·49-s + 5.79·51-s + 9.42·53-s + 5.64·57-s − 13.9·59-s + 10.1·61-s + 3.18·63-s − 8.64·67-s − 4.12·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.20·7-s + 0.333·9-s + 0.0166·11-s + 1.63·13-s + 1.40·17-s + 1.29·19-s + 0.695·21-s − 0.861·23-s + 0.192·27-s + 0.854·29-s − 0.179·31-s + 0.00958·33-s + 0.724·37-s + 0.941·39-s − 0.791·41-s + 1.52·43-s − 0.811·47-s + 0.452·49-s + 0.811·51-s + 1.29·53-s + 0.748·57-s − 1.81·59-s + 1.30·61-s + 0.401·63-s − 1.05·67-s − 0.497·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.027975911\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.027975911\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 - 3.18T + 7T^{2} \) |
| 11 | \( 1 - 0.0550T + 11T^{2} \) |
| 13 | \( 1 - 5.87T + 13T^{2} \) |
| 17 | \( 1 - 5.79T + 17T^{2} \) |
| 19 | \( 1 - 5.64T + 19T^{2} \) |
| 23 | \( 1 + 4.12T + 23T^{2} \) |
| 29 | \( 1 - 4.59T + 29T^{2} \) |
| 37 | \( 1 - 4.40T + 37T^{2} \) |
| 41 | \( 1 + 5.06T + 41T^{2} \) |
| 43 | \( 1 - 10.0T + 43T^{2} \) |
| 47 | \( 1 + 5.56T + 47T^{2} \) |
| 53 | \( 1 - 9.42T + 53T^{2} \) |
| 59 | \( 1 + 13.9T + 59T^{2} \) |
| 61 | \( 1 - 10.1T + 61T^{2} \) |
| 67 | \( 1 + 8.64T + 67T^{2} \) |
| 71 | \( 1 + 12.1T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 + 16.7T + 79T^{2} \) |
| 83 | \( 1 - 7.12T + 83T^{2} \) |
| 89 | \( 1 + 5.61T + 89T^{2} \) |
| 97 | \( 1 + 2.01T + 97T^{2} \) |
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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
−7.71313147296665169503879648053, −7.36839734020855477364498939472, −6.20869105261949554546601961935, −5.69685466802759194500259817233, −4.92513732029982579715686497974, −4.11133133203257349234651900895, −3.46351656285852914630443904171, −2.69504379805480576389363284581, −1.50605808987515664229979942225, −1.11481754364819723296519425117,
1.11481754364819723296519425117, 1.50605808987515664229979942225, 2.69504379805480576389363284581, 3.46351656285852914630443904171, 4.11133133203257349234651900895, 4.92513732029982579715686497974, 5.69685466802759194500259817233, 6.20869105261949554546601961935, 7.36839734020855477364498939472, 7.71313147296665169503879648053
