| L(s) = 1 | − 2.38·2-s + 3.67·4-s − 3.76·5-s − 1.09·7-s − 3.99·8-s + 8.95·10-s + 11-s − 1.78·13-s + 2.60·14-s + 2.16·16-s + 3.04·17-s − 3.42·19-s − 13.8·20-s − 2.38·22-s − 6.13·23-s + 9.14·25-s + 4.24·26-s − 4.01·28-s − 3.50·29-s + 6.38·31-s + 2.83·32-s − 7.25·34-s + 4.10·35-s + 1.67·37-s + 8.16·38-s + 15.0·40-s + 3.11·41-s + ⋯ |
| L(s) = 1 | − 1.68·2-s + 1.83·4-s − 1.68·5-s − 0.412·7-s − 1.41·8-s + 2.83·10-s + 0.301·11-s − 0.494·13-s + 0.694·14-s + 0.540·16-s + 0.738·17-s − 0.786·19-s − 3.09·20-s − 0.507·22-s − 1.27·23-s + 1.82·25-s + 0.832·26-s − 0.758·28-s − 0.649·29-s + 1.14·31-s + 0.500·32-s − 1.24·34-s + 0.693·35-s + 0.275·37-s + 1.32·38-s + 2.37·40-s + 0.486·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 + 2.38T + 2T^{2} \) |
| 5 | \( 1 + 3.76T + 5T^{2} \) |
| 7 | \( 1 + 1.09T + 7T^{2} \) |
| 13 | \( 1 + 1.78T + 13T^{2} \) |
| 17 | \( 1 - 3.04T + 17T^{2} \) |
| 19 | \( 1 + 3.42T + 19T^{2} \) |
| 23 | \( 1 + 6.13T + 23T^{2} \) |
| 29 | \( 1 + 3.50T + 29T^{2} \) |
| 31 | \( 1 - 6.38T + 31T^{2} \) |
| 37 | \( 1 - 1.67T + 37T^{2} \) |
| 41 | \( 1 - 3.11T + 41T^{2} \) |
| 43 | \( 1 + 1.12T + 43T^{2} \) |
| 47 | \( 1 - 0.248T + 47T^{2} \) |
| 53 | \( 1 - 14.3T + 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 67 | \( 1 + 4.06T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 + 0.212T + 73T^{2} \) |
| 79 | \( 1 + 5.71T + 79T^{2} \) |
| 83 | \( 1 - 11.1T + 83T^{2} \) |
| 89 | \( 1 + 8.41T + 89T^{2} \) |
| 97 | \( 1 + 6.22T + 97T^{2} \) |
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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
−7.926853137810987724309235443458, −7.33966541670367772646000209506, −6.70365358222066076434787967009, −5.93981620334178753115171675902, −4.62621488323788966933515916947, −3.93154587819395946485112129120, −3.07231629838619814254086148485, −2.07608490819155204162756584497, −0.832388922203992140079034022956, 0,
0.832388922203992140079034022956, 2.07608490819155204162756584497, 3.07231629838619814254086148485, 3.93154587819395946485112129120, 4.62621488323788966933515916947, 5.93981620334178753115171675902, 6.70365358222066076434787967009, 7.33966541670367772646000209506, 7.926853137810987724309235443458
