L(s) = 1 | − 2-s − 3-s + 4-s + 3·5-s + 6-s − 8-s + 9-s − 3·10-s + 3·11-s − 12-s + 4·13-s − 3·15-s + 16-s − 6·17-s − 18-s − 19-s + 3·20-s − 3·22-s − 6·23-s + 24-s + 4·25-s − 4·26-s − 27-s − 3·29-s + 3·30-s + 31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.34·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.948·10-s + 0.904·11-s − 0.288·12-s + 1.10·13-s − 0.774·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 0.229·19-s + 0.670·20-s − 0.639·22-s − 1.25·23-s + 0.204·24-s + 4/5·25-s − 0.784·26-s − 0.192·27-s − 0.557·29-s + 0.547·30-s + 0.179·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.599138289\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.599138289\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−8.309621658183829826257937569694, −7.35071840686186553314816658779, −6.48075676791858167917121027514, −6.15905108387705756407606659636, −5.64736859507931332446373433013, −4.47538478154447337214871706894, −3.75264772686247995257940946983, −2.35323352832998832939862950765, −1.79770368897729986401577375142, −0.794050740732270172204497719054,
0.794050740732270172204497719054, 1.79770368897729986401577375142, 2.35323352832998832939862950765, 3.75264772686247995257940946983, 4.47538478154447337214871706894, 5.64736859507931332446373433013, 6.15905108387705756407606659636, 6.48075676791858167917121027514, 7.35071840686186553314816658779, 8.309621658183829826257937569694