| L(s) = 1 | − 1.80·2-s − 3.27·3-s + 1.24·4-s + 1.98·5-s + 5.89·6-s + 1.36·8-s + 7.72·9-s − 3.56·10-s − 3.35·11-s − 4.07·12-s − 2.29·13-s − 6.49·15-s − 4.94·16-s − 0.137·17-s − 13.9·18-s + 0.990·19-s + 2.46·20-s + 6.04·22-s − 5.97·23-s − 4.46·24-s − 1.07·25-s + 4.12·26-s − 15.4·27-s − 9.52·29-s + 11.6·30-s − 9.64·31-s + 6.17·32-s + ⋯ |
| L(s) = 1 | − 1.27·2-s − 1.89·3-s + 0.621·4-s + 0.886·5-s + 2.40·6-s + 0.481·8-s + 2.57·9-s − 1.12·10-s − 1.01·11-s − 1.17·12-s − 0.635·13-s − 1.67·15-s − 1.23·16-s − 0.0333·17-s − 3.27·18-s + 0.227·19-s + 0.550·20-s + 1.28·22-s − 1.24·23-s − 0.911·24-s − 0.214·25-s + 0.809·26-s − 2.97·27-s − 1.76·29-s + 2.13·30-s − 1.73·31-s + 1.09·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.07271492430\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07271492430\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 127 | \( 1 - T \) |
| good | 2 | \( 1 + 1.80T + 2T^{2} \) |
| 3 | \( 1 + 3.27T + 3T^{2} \) |
| 5 | \( 1 - 1.98T + 5T^{2} \) |
| 11 | \( 1 + 3.35T + 11T^{2} \) |
| 13 | \( 1 + 2.29T + 13T^{2} \) |
| 17 | \( 1 + 0.137T + 17T^{2} \) |
| 19 | \( 1 - 0.990T + 19T^{2} \) |
| 23 | \( 1 + 5.97T + 23T^{2} \) |
| 29 | \( 1 + 9.52T + 29T^{2} \) |
| 31 | \( 1 + 9.64T + 31T^{2} \) |
| 37 | \( 1 + 10.0T + 37T^{2} \) |
| 41 | \( 1 + 4.24T + 41T^{2} \) |
| 43 | \( 1 - 3.91T + 43T^{2} \) |
| 47 | \( 1 + 3.08T + 47T^{2} \) |
| 53 | \( 1 - 3.23T + 53T^{2} \) |
| 59 | \( 1 + 7.78T + 59T^{2} \) |
| 61 | \( 1 + 3.32T + 61T^{2} \) |
| 67 | \( 1 + 8.34T + 67T^{2} \) |
| 71 | \( 1 + 5.67T + 71T^{2} \) |
| 73 | \( 1 - 7.22T + 73T^{2} \) |
| 79 | \( 1 + 7.90T + 79T^{2} \) |
| 83 | \( 1 - 17.2T + 83T^{2} \) |
| 89 | \( 1 - 5.07T + 89T^{2} \) |
| 97 | \( 1 - 14.4T + 97T^{2} \) |
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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
−7.75812312550305757118649268417, −7.46437963970287778416881016230, −6.70954032204622962986886406972, −5.84498895697064857053481209881, −5.41012049966824705704327807696, −4.81573149945306293466429470019, −3.79185773280474764016450775054, −2.03684858576119134748106963172, −1.63046872042846973597037053772, −0.18836511100548314806223666687,
0.18836511100548314806223666687, 1.63046872042846973597037053772, 2.03684858576119134748106963172, 3.79185773280474764016450775054, 4.81573149945306293466429470019, 5.41012049966824705704327807696, 5.84498895697064857053481209881, 6.70954032204622962986886406972, 7.46437963970287778416881016230, 7.75812312550305757118649268417
