| L(s) = 1 | − 2.45·5-s + 4.94·7-s − 3.80·11-s + 1.47·13-s − 1.18·17-s + 19-s + 3.96·23-s + 1.01·25-s − 7.11·29-s + 2.13·31-s − 12.1·35-s − 5.76·37-s − 0.757·41-s − 3.20·43-s + 4.90·47-s + 17.4·49-s − 6.58·53-s + 9.32·55-s − 1.65·59-s − 8.87·61-s − 3.61·65-s − 6.98·67-s − 13.3·71-s + 10.3·73-s − 18.8·77-s − 8.33·79-s + 13.7·83-s + ⋯ |
| L(s) = 1 | − 1.09·5-s + 1.86·7-s − 1.14·11-s + 0.408·13-s − 0.286·17-s + 0.229·19-s + 0.826·23-s + 0.203·25-s − 1.32·29-s + 0.382·31-s − 2.05·35-s − 0.948·37-s − 0.118·41-s − 0.488·43-s + 0.715·47-s + 2.49·49-s − 0.904·53-s + 1.25·55-s − 0.215·59-s − 1.13·61-s − 0.447·65-s − 0.852·67-s − 1.58·71-s + 1.21·73-s − 2.14·77-s − 0.937·79-s + 1.51·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + 2.45T + 5T^{2} \) |
| 7 | \( 1 - 4.94T + 7T^{2} \) |
| 11 | \( 1 + 3.80T + 11T^{2} \) |
| 13 | \( 1 - 1.47T + 13T^{2} \) |
| 17 | \( 1 + 1.18T + 17T^{2} \) |
| 23 | \( 1 - 3.96T + 23T^{2} \) |
| 29 | \( 1 + 7.11T + 29T^{2} \) |
| 31 | \( 1 - 2.13T + 31T^{2} \) |
| 37 | \( 1 + 5.76T + 37T^{2} \) |
| 41 | \( 1 + 0.757T + 41T^{2} \) |
| 43 | \( 1 + 3.20T + 43T^{2} \) |
| 47 | \( 1 - 4.90T + 47T^{2} \) |
| 53 | \( 1 + 6.58T + 53T^{2} \) |
| 59 | \( 1 + 1.65T + 59T^{2} \) |
| 61 | \( 1 + 8.87T + 61T^{2} \) |
| 67 | \( 1 + 6.98T + 67T^{2} \) |
| 71 | \( 1 + 13.3T + 71T^{2} \) |
| 73 | \( 1 - 10.3T + 73T^{2} \) |
| 79 | \( 1 + 8.33T + 79T^{2} \) |
| 83 | \( 1 - 13.7T + 83T^{2} \) |
| 89 | \( 1 - 8.85T + 89T^{2} \) |
| 97 | \( 1 - 11.1T + 97T^{2} \) |
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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
−7.66223849324257383353545758916, −7.11090439917776975627089844362, −5.95298253066078896035138688621, −5.13919138422186520188879737357, −4.75789198405685273410072551772, −3.98735420126844875424662313316, −3.16535532994092519949011852785, −2.13999642405681780276031405529, −1.28217653280187579856990231594, 0,
1.28217653280187579856990231594, 2.13999642405681780276031405529, 3.16535532994092519949011852785, 3.98735420126844875424662313316, 4.75789198405685273410072551772, 5.13919138422186520188879737357, 5.95298253066078896035138688621, 7.11090439917776975627089844362, 7.66223849324257383353545758916
