| L(s) = 1 | − 1.03·2-s − 0.921·4-s − 2.59·5-s + 2.45·7-s + 3.03·8-s + 2.69·10-s + 11-s − 0.671·13-s − 2.55·14-s − 1.30·16-s − 6.21·17-s + 1.35·19-s + 2.38·20-s − 1.03·22-s + 4.68·23-s + 1.70·25-s + 0.697·26-s − 2.26·28-s − 3.75·29-s + 10.9·31-s − 4.70·32-s + 6.45·34-s − 6.36·35-s − 4.57·37-s − 1.40·38-s − 7.85·40-s − 7.00·41-s + ⋯ |
| L(s) = 1 | − 0.734·2-s − 0.460·4-s − 1.15·5-s + 0.929·7-s + 1.07·8-s + 0.850·10-s + 0.301·11-s − 0.186·13-s − 0.682·14-s − 0.327·16-s − 1.50·17-s + 0.310·19-s + 0.533·20-s − 0.221·22-s + 0.977·23-s + 0.341·25-s + 0.136·26-s − 0.427·28-s − 0.697·29-s + 1.96·31-s − 0.832·32-s + 1.10·34-s − 1.07·35-s − 0.751·37-s − 0.228·38-s − 1.24·40-s − 1.09·41-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 | \( 1 + 1.03T + 2T^{2} \) |
| 5 | \( 1 + 2.59T + 5T^{2} \) |
| 7 | \( 1 - 2.45T + 7T^{2} \) |
| 13 | \( 1 + 0.671T + 13T^{2} \) |
| 17 | \( 1 + 6.21T + 17T^{2} \) |
| 19 | \( 1 - 1.35T + 19T^{2} \) |
| 23 | \( 1 - 4.68T + 23T^{2} \) |
| 29 | \( 1 + 3.75T + 29T^{2} \) |
| 31 | \( 1 - 10.9T + 31T^{2} \) |
| 37 | \( 1 + 4.57T + 37T^{2} \) |
| 41 | \( 1 + 7.00T + 41T^{2} \) |
| 43 | \( 1 + 3.15T + 43T^{2} \) |
| 47 | \( 1 + 0.129T + 47T^{2} \) |
| 53 | \( 1 - 5.96T + 53T^{2} \) |
| 59 | \( 1 - 5.85T + 59T^{2} \) |
| 67 | \( 1 - 10.8T + 67T^{2} \) |
| 71 | \( 1 + 7.40T + 71T^{2} \) |
| 73 | \( 1 + 1.18T + 73T^{2} \) |
| 79 | \( 1 + 9.52T + 79T^{2} \) |
| 83 | \( 1 + 8.87T + 83T^{2} \) |
| 89 | \( 1 - 14.7T + 89T^{2} \) |
| 97 | \( 1 - 9.17T + 97T^{2} \) |
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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
−7.921084861394463086845893239056, −7.19544541503205287839317018593, −6.67070658484509347884540601357, −5.33784956490582888304334358672, −4.65338954418253962134012222440, −4.24001250926675701260905962540, −3.32968032732332091278351613779, −2.07935430902680508118757543321, −1.05965139833224351303061115140, 0,
1.05965139833224351303061115140, 2.07935430902680508118757543321, 3.32968032732332091278351613779, 4.24001250926675701260905962540, 4.65338954418253962134012222440, 5.33784956490582888304334358672, 6.67070658484509347884540601357, 7.19544541503205287839317018593, 7.921084861394463086845893239056
