| L(s) = 1 | − 3-s + 3.58·5-s − 1.47·7-s + 9-s − 2.22·11-s + 3.79·13-s − 3.58·15-s − 4.21·17-s − 2.92·19-s + 1.47·21-s − 1.45·23-s + 7.85·25-s − 27-s − 8.55·29-s + 7.83·31-s + 2.22·33-s − 5.30·35-s − 2.57·37-s − 3.79·39-s − 4.27·41-s + 4.13·43-s + 3.58·45-s − 3.07·47-s − 4.80·49-s + 4.21·51-s + 2.67·53-s − 7.97·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.60·5-s − 0.559·7-s + 0.333·9-s − 0.670·11-s + 1.05·13-s − 0.925·15-s − 1.02·17-s − 0.671·19-s + 0.322·21-s − 0.303·23-s + 1.57·25-s − 0.192·27-s − 1.58·29-s + 1.40·31-s + 0.386·33-s − 0.896·35-s − 0.423·37-s − 0.607·39-s − 0.668·41-s + 0.630·43-s + 0.534·45-s − 0.448·47-s − 0.687·49-s + 0.589·51-s + 0.367·53-s − 1.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 251 | \( 1 - T \) |
| good | 5 | \( 1 - 3.58T + 5T^{2} \) |
| 7 | \( 1 + 1.47T + 7T^{2} \) |
| 11 | \( 1 + 2.22T + 11T^{2} \) |
| 13 | \( 1 - 3.79T + 13T^{2} \) |
| 17 | \( 1 + 4.21T + 17T^{2} \) |
| 19 | \( 1 + 2.92T + 19T^{2} \) |
| 23 | \( 1 + 1.45T + 23T^{2} \) |
| 29 | \( 1 + 8.55T + 29T^{2} \) |
| 31 | \( 1 - 7.83T + 31T^{2} \) |
| 37 | \( 1 + 2.57T + 37T^{2} \) |
| 41 | \( 1 + 4.27T + 41T^{2} \) |
| 43 | \( 1 - 4.13T + 43T^{2} \) |
| 47 | \( 1 + 3.07T + 47T^{2} \) |
| 53 | \( 1 - 2.67T + 53T^{2} \) |
| 59 | \( 1 + 12.0T + 59T^{2} \) |
| 61 | \( 1 - 3.15T + 61T^{2} \) |
| 67 | \( 1 + 13.4T + 67T^{2} \) |
| 71 | \( 1 + 13.3T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 + 6.01T + 79T^{2} \) |
| 83 | \( 1 - 3.45T + 83T^{2} \) |
| 89 | \( 1 - 15.3T + 89T^{2} \) |
| 97 | \( 1 + 9.36T + 97T^{2} \) |
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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
−7.65523454754878171377508896178, −6.63023926375417515769889248485, −6.27789962195029697010543078919, −5.75452851364399209961093155536, −5.01009825784013786268675225979, −4.17210301086003678748672641521, −3.09085058045304476516043011451, −2.18808798614823486711201836937, −1.44930702281622652841593416927, 0,
1.44930702281622652841593416927, 2.18808798614823486711201836937, 3.09085058045304476516043011451, 4.17210301086003678748672641521, 5.01009825784013786268675225979, 5.75452851364399209961093155536, 6.27789962195029697010543078919, 6.63023926375417515769889248485, 7.65523454754878171377508896178
