| L(s) = 1 | + 0.500·2-s − 1.74·4-s − 0.0237·7-s − 1.87·8-s − 3.58·11-s − 3.77·13-s − 0.0119·14-s + 2.55·16-s − 3.62·17-s − 2.43·19-s − 1.79·22-s + 1.71·23-s − 1.89·26-s + 0.0416·28-s − 3.85·29-s − 6.00·31-s + 5.03·32-s − 1.81·34-s + 0.369·37-s − 1.21·38-s + 7.80·41-s − 0.174·43-s + 6.26·44-s + 0.859·46-s + 7.81·47-s − 6.99·49-s + 6.60·52-s + ⋯ |
| L(s) = 1 | + 0.354·2-s − 0.874·4-s − 0.00899·7-s − 0.664·8-s − 1.08·11-s − 1.04·13-s − 0.00318·14-s + 0.639·16-s − 0.878·17-s − 0.557·19-s − 0.382·22-s + 0.357·23-s − 0.370·26-s + 0.00786·28-s − 0.716·29-s − 1.07·31-s + 0.890·32-s − 0.311·34-s + 0.0607·37-s − 0.197·38-s + 1.21·41-s − 0.0266·43-s + 0.944·44-s + 0.126·46-s + 1.13·47-s − 0.999·49-s + 0.915·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8327880515\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8327880515\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 0.500T + 2T^{2} \) |
| 7 | \( 1 + 0.0237T + 7T^{2} \) |
| 11 | \( 1 + 3.58T + 11T^{2} \) |
| 13 | \( 1 + 3.77T + 13T^{2} \) |
| 17 | \( 1 + 3.62T + 17T^{2} \) |
| 19 | \( 1 + 2.43T + 19T^{2} \) |
| 23 | \( 1 - 1.71T + 23T^{2} \) |
| 29 | \( 1 + 3.85T + 29T^{2} \) |
| 31 | \( 1 + 6.00T + 31T^{2} \) |
| 37 | \( 1 - 0.369T + 37T^{2} \) |
| 41 | \( 1 - 7.80T + 41T^{2} \) |
| 43 | \( 1 + 0.174T + 43T^{2} \) |
| 47 | \( 1 - 7.81T + 47T^{2} \) |
| 53 | \( 1 + 8.97T + 53T^{2} \) |
| 59 | \( 1 - 4.45T + 59T^{2} \) |
| 61 | \( 1 - 9.21T + 61T^{2} \) |
| 67 | \( 1 - 4.47T + 67T^{2} \) |
| 71 | \( 1 - 9.69T + 71T^{2} \) |
| 73 | \( 1 - 3.95T + 73T^{2} \) |
| 79 | \( 1 + 9.68T + 79T^{2} \) |
| 83 | \( 1 + 8.95T + 83T^{2} \) |
| 89 | \( 1 - 17.0T + 89T^{2} \) |
| 97 | \( 1 + 2.76T + 97T^{2} \) |
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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
−8.119447697861444751017943953260, −7.50139895816720452521767396833, −6.71079414734899983368612810195, −5.73833061212597076756995005294, −5.19044243394389204474779162118, −4.54569361574381229245862162548, −3.82944927098409309395557129425, −2.83898770943768097868818671504, −2.07456571866926641126961132482, −0.43870647122646917293189604294,
0.43870647122646917293189604294, 2.07456571866926641126961132482, 2.83898770943768097868818671504, 3.82944927098409309395557129425, 4.54569361574381229245862162548, 5.19044243394389204474779162118, 5.73833061212597076756995005294, 6.71079414734899983368612810195, 7.50139895816720452521767396833, 8.119447697861444751017943953260
