| L(s) = 1 | − 4·11-s + 13-s + 3·17-s − 6·19-s − 3·23-s − 3·29-s + 9·31-s − 6·37-s − 5·41-s + 9·43-s − 6·47-s + 9·53-s − 13·59-s + 7·61-s + 8·67-s + 16·71-s − 8·73-s − 12·79-s − 15·83-s − 14·89-s − 8·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.20·11-s + 0.277·13-s + 0.727·17-s − 1.37·19-s − 0.625·23-s − 0.557·29-s + 1.61·31-s − 0.986·37-s − 0.780·41-s + 1.37·43-s − 0.875·47-s + 1.23·53-s − 1.69·59-s + 0.896·61-s + 0.977·67-s + 1.89·71-s − 0.936·73-s − 1.35·79-s − 1.64·83-s − 1.48·89-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9718670453\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9718670453\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 13 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−13.04634416199991, −12.78036597592092, −12.30176344398110, −11.82732922841687, −11.21122322168046, −10.80558375873822, −10.30247210947153, −9.963387111698537, −9.521734502897889, −8.648631988669476, −8.388544316730885, −8.043665621837091, −7.379065919129187, −6.936337305833777, −6.289418745416019, −5.858985682262547, −5.315393954251039, −4.847765450547637, −4.141253415764074, −3.787412677757676, −2.929230291718888, −2.572656865792579, −1.889471846793335, −1.209863691541425, −0.2866964890558333,
0.2866964890558333, 1.209863691541425, 1.889471846793335, 2.572656865792579, 2.929230291718888, 3.787412677757676, 4.141253415764074, 4.847765450547637, 5.315393954251039, 5.858985682262547, 6.289418745416019, 6.936337305833777, 7.379065919129187, 8.043665621837091, 8.388544316730885, 8.648631988669476, 9.521734502897889, 9.963387111698537, 10.30247210947153, 10.80558375873822, 11.21122322168046, 11.82732922841687, 12.30176344398110, 12.78036597592092, 13.04634416199991
