| L(s) = 1 | + 0.806·3-s + 2.15·7-s − 2.35·9-s − 0.387·11-s − 0.962·13-s + 1.61·17-s + 6.31·19-s + 1.73·21-s − 6.15·23-s − 4.31·27-s − 2·29-s + 9.92·31-s − 0.312·33-s + 6.57·37-s − 0.775·39-s + 4.57·41-s + 11.5·43-s − 4.54·47-s − 2.35·49-s + 1.29·51-s + 8.96·53-s + 5.08·57-s − 6.31·59-s − 0.261·61-s − 5.06·63-s + 9.89·67-s − 4.96·69-s + ⋯ |
| L(s) = 1 | + 0.465·3-s + 0.815·7-s − 0.783·9-s − 0.116·11-s − 0.266·13-s + 0.390·17-s + 1.44·19-s + 0.379·21-s − 1.28·23-s − 0.829·27-s − 0.371·29-s + 1.78·31-s − 0.0544·33-s + 1.08·37-s − 0.124·39-s + 0.714·41-s + 1.75·43-s − 0.662·47-s − 0.335·49-s + 0.181·51-s + 1.23·53-s + 0.673·57-s − 0.821·59-s − 0.0335·61-s − 0.638·63-s + 1.20·67-s − 0.597·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.304090025\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.304090025\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 0.806T + 3T^{2} \) |
| 7 | \( 1 - 2.15T + 7T^{2} \) |
| 11 | \( 1 + 0.387T + 11T^{2} \) |
| 13 | \( 1 + 0.962T + 13T^{2} \) |
| 17 | \( 1 - 1.61T + 17T^{2} \) |
| 19 | \( 1 - 6.31T + 19T^{2} \) |
| 23 | \( 1 + 6.15T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 9.92T + 31T^{2} \) |
| 37 | \( 1 - 6.57T + 37T^{2} \) |
| 41 | \( 1 - 4.57T + 41T^{2} \) |
| 43 | \( 1 - 11.5T + 43T^{2} \) |
| 47 | \( 1 + 4.54T + 47T^{2} \) |
| 53 | \( 1 - 8.96T + 53T^{2} \) |
| 59 | \( 1 + 6.31T + 59T^{2} \) |
| 61 | \( 1 + 0.261T + 61T^{2} \) |
| 67 | \( 1 - 9.89T + 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 - 13.0T + 73T^{2} \) |
| 79 | \( 1 + 1.92T + 79T^{2} \) |
| 83 | \( 1 - 2.88T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 - 9.61T + 97T^{2} \) |
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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
−8.548477089140048608634280951219, −7.80959392968645876183728114889, −7.60550200502344598342553327369, −6.28264717680533170321999305362, −5.62828377538297839016106892659, −4.83451532489689195433874734414, −3.93371884234546285162477885493, −2.94431944730512403354088444844, −2.20001635592051120872546725096, −0.910953786201455827732466803708,
0.910953786201455827732466803708, 2.20001635592051120872546725096, 2.94431944730512403354088444844, 3.93371884234546285162477885493, 4.83451532489689195433874734414, 5.62828377538297839016106892659, 6.28264717680533170321999305362, 7.60550200502344598342553327369, 7.80959392968645876183728114889, 8.548477089140048608634280951219
