| L(s) = 1 | − 2.32·3-s + 3.89·5-s + 0.0996·7-s + 2.39·9-s − 5.90·11-s − 0.114·13-s − 9.04·15-s − 0.598·17-s − 3.75·19-s − 0.231·21-s + 8.45·23-s + 10.1·25-s + 1.40·27-s − 4.82·29-s − 5.02·31-s + 13.7·33-s + 0.388·35-s + 4.30·37-s + 0.265·39-s − 1.72·41-s + 7.69·43-s + 9.33·45-s + 3.57·47-s − 6.99·49-s + 1.39·51-s + 8.09·53-s − 23.0·55-s + ⋯ |
| L(s) = 1 | − 1.34·3-s + 1.74·5-s + 0.0376·7-s + 0.798·9-s − 1.78·11-s − 0.0316·13-s − 2.33·15-s − 0.145·17-s − 0.862·19-s − 0.0505·21-s + 1.76·23-s + 2.03·25-s + 0.270·27-s − 0.896·29-s − 0.901·31-s + 2.38·33-s + 0.0656·35-s + 0.708·37-s + 0.0424·39-s − 0.270·41-s + 1.17·43-s + 1.39·45-s + 0.521·47-s − 0.998·49-s + 0.194·51-s + 1.11·53-s − 3.10·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 751 | \( 1 + T \) |
| good | 3 | \( 1 + 2.32T + 3T^{2} \) |
| 5 | \( 1 - 3.89T + 5T^{2} \) |
| 7 | \( 1 - 0.0996T + 7T^{2} \) |
| 11 | \( 1 + 5.90T + 11T^{2} \) |
| 13 | \( 1 + 0.114T + 13T^{2} \) |
| 17 | \( 1 + 0.598T + 17T^{2} \) |
| 19 | \( 1 + 3.75T + 19T^{2} \) |
| 23 | \( 1 - 8.45T + 23T^{2} \) |
| 29 | \( 1 + 4.82T + 29T^{2} \) |
| 31 | \( 1 + 5.02T + 31T^{2} \) |
| 37 | \( 1 - 4.30T + 37T^{2} \) |
| 41 | \( 1 + 1.72T + 41T^{2} \) |
| 43 | \( 1 - 7.69T + 43T^{2} \) |
| 47 | \( 1 - 3.57T + 47T^{2} \) |
| 53 | \( 1 - 8.09T + 53T^{2} \) |
| 59 | \( 1 + 8.16T + 59T^{2} \) |
| 61 | \( 1 - 12.5T + 61T^{2} \) |
| 67 | \( 1 - 0.648T + 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 + 16.5T + 73T^{2} \) |
| 79 | \( 1 + 3.11T + 79T^{2} \) |
| 83 | \( 1 + 12.1T + 83T^{2} \) |
| 89 | \( 1 + 4.53T + 89T^{2} \) |
| 97 | \( 1 + 12.9T + 97T^{2} \) |
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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
−7.43186950972058296707814469175, −6.89565075539963675510064971119, −5.98398774382589369938863284693, −5.64440925932813082283833781174, −5.15419406183325686667051281596, −4.46835448455907992664960744690, −2.91536004726570315944503871953, −2.29227200539071883718182257369, −1.24786530704091697553979774520, 0,
1.24786530704091697553979774520, 2.29227200539071883718182257369, 2.91536004726570315944503871953, 4.46835448455907992664960744690, 5.15419406183325686667051281596, 5.64440925932813082283833781174, 5.98398774382589369938863284693, 6.89565075539963675510064971119, 7.43186950972058296707814469175
