L(s) = 1 | − 2·3-s + 9-s + 11-s + 4·13-s + 7·17-s + 4·19-s + 9·23-s + 4·27-s − 10·29-s − 5·31-s − 2·33-s + 10·37-s − 8·39-s − 41-s + 12·43-s − 9·47-s − 14·51-s − 4·53-s − 8·57-s − 6·59-s + 6·61-s − 8·67-s − 18·69-s − 11·71-s + 2·73-s + 13·79-s − 11·81-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 1/3·9-s + 0.301·11-s + 1.10·13-s + 1.69·17-s + 0.917·19-s + 1.87·23-s + 0.769·27-s − 1.85·29-s − 0.898·31-s − 0.348·33-s + 1.64·37-s − 1.28·39-s − 0.156·41-s + 1.82·43-s − 1.31·47-s − 1.96·51-s − 0.549·53-s − 1.05·57-s − 0.781·59-s + 0.768·61-s − 0.977·67-s − 2.16·69-s − 1.30·71-s + 0.234·73-s + 1.46·79-s − 1.22·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 107800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 107800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.024877371\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.024877371\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 11 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 17 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−13.56930921650523, −13.03494695678730, −12.70879190248005, −12.17788294015969, −11.56247217859144, −11.26630412938827, −10.90314759623485, −10.48789615482889, −9.636270082531077, −9.346891020080565, −8.910614022552320, −8.048434226573168, −7.638250847792392, −7.153576701070231, −6.513614686795574, −5.952031380994221, −5.531259503958114, −5.279450834227533, −4.511819441463824, −3.837599389706299, −3.248890012043120, −2.811235402215253, −1.581349915370090, −1.156225522774129, −0.5571620045164575,
0.5571620045164575, 1.156225522774129, 1.581349915370090, 2.811235402215253, 3.248890012043120, 3.837599389706299, 4.511819441463824, 5.279450834227533, 5.531259503958114, 5.952031380994221, 6.513614686795574, 7.153576701070231, 7.638250847792392, 8.048434226573168, 8.910614022552320, 9.346891020080565, 9.636270082531077, 10.48789615482889, 10.90314759623485, 11.26630412938827, 11.56247217859144, 12.17788294015969, 12.70879190248005, 13.03494695678730, 13.56930921650523