L(s) = 1 | − 0.369·2-s + 2.05·3-s − 1.86·4-s − 2.66·5-s − 0.761·6-s − 0.747·7-s + 1.42·8-s + 1.24·9-s + 0.983·10-s + 1.65·11-s − 3.83·12-s + 2.02·13-s + 0.276·14-s − 5.48·15-s + 3.19·16-s + 4.36·17-s − 0.459·18-s + 19-s + 4.96·20-s − 1.53·21-s − 0.611·22-s + 0.850·23-s + 2.94·24-s + 2.08·25-s − 0.746·26-s − 3.62·27-s + 1.39·28-s + ⋯ |
L(s) = 1 | − 0.261·2-s + 1.18·3-s − 0.931·4-s − 1.19·5-s − 0.310·6-s − 0.282·7-s + 0.504·8-s + 0.414·9-s + 0.311·10-s + 0.498·11-s − 1.10·12-s + 0.560·13-s + 0.0738·14-s − 1.41·15-s + 0.799·16-s + 1.05·17-s − 0.108·18-s + 0.229·19-s + 1.10·20-s − 0.335·21-s − 0.130·22-s + 0.177·23-s + 0.600·24-s + 0.417·25-s − 0.146·26-s − 0.696·27-s + 0.263·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.423217144\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.423217144\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 19 | \( 1 - T \) |
| 317 | \( 1 + T \) |
good | 2 | \( 1 + 0.369T + 2T^{2} \) |
| 3 | \( 1 - 2.05T + 3T^{2} \) |
| 5 | \( 1 + 2.66T + 5T^{2} \) |
| 7 | \( 1 + 0.747T + 7T^{2} \) |
| 11 | \( 1 - 1.65T + 11T^{2} \) |
| 13 | \( 1 - 2.02T + 13T^{2} \) |
| 17 | \( 1 - 4.36T + 17T^{2} \) |
| 23 | \( 1 - 0.850T + 23T^{2} \) |
| 29 | \( 1 + 7.02T + 29T^{2} \) |
| 31 | \( 1 + 6.48T + 31T^{2} \) |
| 37 | \( 1 - 4.42T + 37T^{2} \) |
| 41 | \( 1 + 5.33T + 41T^{2} \) |
| 43 | \( 1 - 0.593T + 43T^{2} \) |
| 47 | \( 1 - 8.86T + 47T^{2} \) |
| 53 | \( 1 + 2.90T + 53T^{2} \) |
| 59 | \( 1 - 5.19T + 59T^{2} \) |
| 61 | \( 1 - 5.03T + 61T^{2} \) |
| 67 | \( 1 - 5.44T + 67T^{2} \) |
| 71 | \( 1 + 9.72T + 71T^{2} \) |
| 73 | \( 1 - 9.87T + 73T^{2} \) |
| 79 | \( 1 - 11.7T + 79T^{2} \) |
| 83 | \( 1 - 15.6T + 83T^{2} \) |
| 89 | \( 1 + 5.44T + 89T^{2} \) |
| 97 | \( 1 + 7.02T + 97T^{2} \) |
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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
−8.051316377964515667884287330104, −7.73958626151440478824782623077, −7.04247304923254720523416404434, −5.86054314371594541165258968816, −5.10477072280991656101222146977, −3.93935561587391507448865651663, −3.75748834973306218588789870655, −3.12390698180174373271013710730, −1.79404019670314178856676779789, −0.62764191721219461665686621555,
0.62764191721219461665686621555, 1.79404019670314178856676779789, 3.12390698180174373271013710730, 3.75748834973306218588789870655, 3.93935561587391507448865651663, 5.10477072280991656101222146977, 5.86054314371594541165258968816, 7.04247304923254720523416404434, 7.73958626151440478824782623077, 8.051316377964515667884287330104