L(s) = 1 | − 2·4-s + 4·16-s − 19-s − 5·25-s − 7·31-s − 11·37-s + 8·43-s + 61-s − 8·64-s − 11·67-s − 10·73-s + 2·76-s − 13·79-s − 19·97-s + 10·100-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
L(s) = 1 | − 4-s + 16-s − 0.229·19-s − 25-s − 1.25·31-s − 1.80·37-s + 1.21·43-s + 0.128·61-s − 64-s − 1.34·67-s − 1.17·73-s + 0.229·76-s − 1.46·79-s − 1.92·97-s + 100-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5219183603\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5219183603\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 19 T + p T^{2} \) |
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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
−14.04358971460105, −13.54024357337632, −13.19894170031243, −12.55706431217002, −12.21206717282148, −11.68093048002791, −10.91994649282941, −10.57322261557240, −9.971996925167838, −9.480490920589601, −9.001772142967933, −8.560949298905203, −8.027869925469345, −7.403180736234225, −6.993016638902868, −6.155396788543186, −5.526679930756300, −5.347826967087567, −4.363777379246020, −4.146006997395092, −3.439623215545755, −2.837836702661041, −1.889882480826932, −1.322589216409674, −0.2468997505725353,
0.2468997505725353, 1.322589216409674, 1.889882480826932, 2.837836702661041, 3.439623215545755, 4.146006997395092, 4.363777379246020, 5.347826967087567, 5.526679930756300, 6.155396788543186, 6.993016638902868, 7.403180736234225, 8.027869925469345, 8.560949298905203, 9.001772142967933, 9.480490920589601, 9.971996925167838, 10.57322261557240, 10.91994649282941, 11.68093048002791, 12.21206717282148, 12.55706431217002, 13.19894170031243, 13.54024357337632, 14.04358971460105