| L(s) = 1 | − 0.0822·2-s + 1.69·3-s − 1.99·4-s − 0.139·6-s + 7-s + 0.328·8-s − 0.117·9-s − 6.60·11-s − 3.38·12-s − 0.246·13-s − 0.0822·14-s + 3.95·16-s − 3.13·17-s + 0.00963·18-s + 3.09·19-s + 1.69·21-s + 0.542·22-s + 23-s + 0.557·24-s + 0.0202·26-s − 5.29·27-s − 1.99·28-s + 1.93·29-s + 11.0·31-s − 0.982·32-s − 11.2·33-s + 0.257·34-s + ⋯ |
| L(s) = 1 | − 0.0581·2-s + 0.980·3-s − 0.996·4-s − 0.0570·6-s + 0.377·7-s + 0.116·8-s − 0.0390·9-s − 1.99·11-s − 0.976·12-s − 0.0682·13-s − 0.0219·14-s + 0.989·16-s − 0.759·17-s + 0.00227·18-s + 0.709·19-s + 0.370·21-s + 0.115·22-s + 0.208·23-s + 0.113·24-s + 0.00396·26-s − 1.01·27-s − 0.376·28-s + 0.360·29-s + 1.97·31-s − 0.173·32-s − 1.95·33-s + 0.0441·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.577374116\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.577374116\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 + 0.0822T + 2T^{2} \) |
| 3 | \( 1 - 1.69T + 3T^{2} \) |
| 11 | \( 1 + 6.60T + 11T^{2} \) |
| 13 | \( 1 + 0.246T + 13T^{2} \) |
| 17 | \( 1 + 3.13T + 17T^{2} \) |
| 19 | \( 1 - 3.09T + 19T^{2} \) |
| 29 | \( 1 - 1.93T + 29T^{2} \) |
| 31 | \( 1 - 11.0T + 31T^{2} \) |
| 37 | \( 1 + 3.70T + 37T^{2} \) |
| 41 | \( 1 + 1.36T + 41T^{2} \) |
| 43 | \( 1 - 8.92T + 43T^{2} \) |
| 47 | \( 1 + 3.34T + 47T^{2} \) |
| 53 | \( 1 - 6.98T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 + 2.35T + 61T^{2} \) |
| 67 | \( 1 - 9.34T + 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 - 11.5T + 73T^{2} \) |
| 79 | \( 1 - 8.34T + 79T^{2} \) |
| 83 | \( 1 + 14.8T + 83T^{2} \) |
| 89 | \( 1 - 15.2T + 89T^{2} \) |
| 97 | \( 1 - 7.03T + 97T^{2} \) |
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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
−8.593329248859878681262347242644, −7.83996980232396575009477305644, −7.42965100532061261604557118144, −6.11843244134863232700812678767, −5.18019800068587377320927891724, −4.79642727687554024192981496595, −3.76767993346666689251469450344, −2.87227058772071637400111780446, −2.26111261471750003723890686898, −0.67536264485154824121101761773,
0.67536264485154824121101761773, 2.26111261471750003723890686898, 2.87227058772071637400111780446, 3.76767993346666689251469450344, 4.79642727687554024192981496595, 5.18019800068587377320927891724, 6.11843244134863232700812678767, 7.42965100532061261604557118144, 7.83996980232396575009477305644, 8.593329248859878681262347242644
