L(s) = 1 | − 3-s − 5-s + 9-s + 4·11-s − 6·13-s + 15-s − 6·17-s − 4·19-s − 23-s + 25-s − 27-s − 6·29-s + 8·31-s − 4·33-s + 6·37-s + 6·39-s + 10·41-s − 4·43-s − 45-s + 8·47-s − 7·49-s + 6·51-s − 14·53-s − 4·55-s + 4·57-s + 10·61-s + 6·65-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s + 1.20·11-s − 1.66·13-s + 0.258·15-s − 1.45·17-s − 0.917·19-s − 0.208·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s + 1.43·31-s − 0.696·33-s + 0.986·37-s + 0.960·39-s + 1.56·41-s − 0.609·43-s − 0.149·45-s + 1.16·47-s − 49-s + 0.840·51-s − 1.92·53-s − 0.539·55-s + 0.529·57-s + 1.28·61-s + 0.744·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9402097950\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9402097950\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−7.992566543121532563262826668327, −7.43291615840038730872869435042, −6.49815120763258671355977199510, −6.34022241526993833619778949486, −5.11042590414889888649553440704, −4.43290229599951863767613129043, −4.01025450946095047079087522600, −2.71039816028535505536496619711, −1.88556668462988862295315555065, −0.51912909136065485186715757218,
0.51912909136065485186715757218, 1.88556668462988862295315555065, 2.71039816028535505536496619711, 4.01025450946095047079087522600, 4.43290229599951863767613129043, 5.11042590414889888649553440704, 6.34022241526993833619778949486, 6.49815120763258671355977199510, 7.43291615840038730872869435042, 7.992566543121532563262826668327