L(s) = 1 | − 3-s + 7-s + 9-s + 2·11-s − 2·13-s − 8·17-s + 2·19-s − 21-s − 27-s − 6·29-s − 6·31-s − 2·33-s − 8·37-s + 2·39-s + 6·41-s + 8·43-s + 4·47-s + 49-s + 8·51-s + 2·53-s − 2·57-s + 8·59-s + 10·61-s + 63-s − 12·67-s + 14·71-s + 10·73-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.603·11-s − 0.554·13-s − 1.94·17-s + 0.458·19-s − 0.218·21-s − 0.192·27-s − 1.11·29-s − 1.07·31-s − 0.348·33-s − 1.31·37-s + 0.320·39-s + 0.937·41-s + 1.21·43-s + 0.583·47-s + 1/7·49-s + 1.12·51-s + 0.274·53-s − 0.264·57-s + 1.04·59-s + 1.28·61-s + 0.125·63-s − 1.46·67-s + 1.66·71-s + 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.321653151\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.321653151\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−7.61330604544962861987041849812, −7.04021555185519770291354341750, −6.52644159275399508941499285396, −5.60348518640638242355745812656, −5.12391047641036388377696544196, −4.22805707910848274048786457590, −3.76075417711346364817964521714, −2.43956727371447946081851846219, −1.80232360887760739051525204114, −0.57234222032455116885962028733,
0.57234222032455116885962028733, 1.80232360887760739051525204114, 2.43956727371447946081851846219, 3.76075417711346364817964521714, 4.22805707910848274048786457590, 5.12391047641036388377696544196, 5.60348518640638242355745812656, 6.52644159275399508941499285396, 7.04021555185519770291354341750, 7.61330604544962861987041849812