L(s) = 1 | − 3-s − 2·7-s + 9-s + 11-s + 5·13-s + 2·17-s + 2·19-s + 2·21-s + 23-s − 27-s − 10·29-s + 31-s − 33-s − 2·37-s − 5·39-s − 9·41-s + 43-s − 13·47-s − 3·49-s − 2·51-s − 53-s − 2·57-s + 7·59-s − 8·61-s − 2·63-s − 4·67-s − 69-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s + 0.301·11-s + 1.38·13-s + 0.485·17-s + 0.458·19-s + 0.436·21-s + 0.208·23-s − 0.192·27-s − 1.85·29-s + 0.179·31-s − 0.174·33-s − 0.328·37-s − 0.800·39-s − 1.40·41-s + 0.152·43-s − 1.89·47-s − 3/7·49-s − 0.280·51-s − 0.137·53-s − 0.264·57-s + 0.911·59-s − 1.02·61-s − 0.251·63-s − 0.488·67-s − 0.120·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.295900880\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.295900880\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 43 | \( 1 - T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 + 13 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
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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
−14.65829170166265, −13.80116351586214, −13.34139768416222, −13.03625731504211, −12.50496976225062, −11.72076455186116, −11.54090917920505, −10.94116328903654, −10.34749069531413, −9.884832964760508, −9.316667695523072, −8.845668958354792, −8.193552988990052, −7.573764417568936, −6.972306436041408, −6.406586129701975, −6.008908393178370, −5.427152890768058, −4.842325337957458, −4.020477356000013, −3.404915492198689, −3.148229697558415, −1.858522759469608, −1.387640649414383, −0.4268741492363611,
0.4268741492363611, 1.387640649414383, 1.858522759469608, 3.148229697558415, 3.404915492198689, 4.020477356000013, 4.842325337957458, 5.427152890768058, 6.008908393178370, 6.406586129701975, 6.972306436041408, 7.573764417568936, 8.193552988990052, 8.845668958354792, 9.316667695523072, 9.884832964760508, 10.34749069531413, 10.94116328903654, 11.54090917920505, 11.72076455186116, 12.50496976225062, 13.03625731504211, 13.34139768416222, 13.80116351586214, 14.65829170166265