| L(s) = 1 | + 1.56·3-s − 3.56·5-s − 3.56·7-s − 0.561·9-s − 2·11-s − 13-s − 5.56·15-s + 3.56·17-s − 6·19-s − 5.56·21-s + 7.68·25-s − 5.56·27-s + 8.24·29-s + 1.12·31-s − 3.12·33-s + 12.6·35-s + 2.68·37-s − 1.56·39-s − 1.12·41-s − 11.8·43-s + 2·45-s + 10.6·47-s + 5.68·49-s + 5.56·51-s − 13.1·53-s + 7.12·55-s − 9.36·57-s + ⋯ |
| L(s) = 1 | + 0.901·3-s − 1.59·5-s − 1.34·7-s − 0.187·9-s − 0.603·11-s − 0.277·13-s − 1.43·15-s + 0.863·17-s − 1.37·19-s − 1.21·21-s + 1.53·25-s − 1.07·27-s + 1.53·29-s + 0.201·31-s − 0.543·33-s + 2.14·35-s + 0.441·37-s − 0.250·39-s − 0.175·41-s − 1.80·43-s + 0.298·45-s + 1.55·47-s + 0.812·49-s + 0.778·51-s − 1.80·53-s + 0.960·55-s − 1.24·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 1.56T + 3T^{2} \) |
| 5 | \( 1 + 3.56T + 5T^{2} \) |
| 7 | \( 1 + 3.56T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 17 | \( 1 - 3.56T + 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 8.24T + 29T^{2} \) |
| 31 | \( 1 - 1.12T + 31T^{2} \) |
| 37 | \( 1 - 2.68T + 37T^{2} \) |
| 41 | \( 1 + 1.12T + 41T^{2} \) |
| 43 | \( 1 + 11.8T + 43T^{2} \) |
| 47 | \( 1 - 10.6T + 47T^{2} \) |
| 53 | \( 1 + 13.1T + 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 + 11.3T + 61T^{2} \) |
| 67 | \( 1 + 6T + 67T^{2} \) |
| 71 | \( 1 + 10.6T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 + 7.36T + 83T^{2} \) |
| 89 | \( 1 - 8.24T + 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−10.67565088589767443246463949915, −9.817551941800209256085620937533, −8.686642160719022342144432527557, −8.113845820974984917274798486075, −7.25185454608095247488917996700, −6.16947279325834065838914490210, −4.55268808714137228837513162197, −3.45306201677728978971783322968, −2.80712463094253039902674874006, 0,
2.80712463094253039902674874006, 3.45306201677728978971783322968, 4.55268808714137228837513162197, 6.16947279325834065838914490210, 7.25185454608095247488917996700, 8.113845820974984917274798486075, 8.686642160719022342144432527557, 9.817551941800209256085620937533, 10.67565088589767443246463949915
