L(s) = 1 | − 2-s − 2.51·3-s + 4-s − 0.511·5-s + 2.51·6-s + 2.88·7-s − 8-s + 3.30·9-s + 0.511·10-s − 5.38·11-s − 2.51·12-s + 3.75·13-s − 2.88·14-s + 1.28·15-s + 16-s − 6.96·17-s − 3.30·18-s − 8.21·19-s − 0.511·20-s − 7.24·21-s + 5.38·22-s − 23-s + 2.51·24-s − 4.73·25-s − 3.75·26-s − 0.760·27-s + 2.88·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.44·3-s + 0.5·4-s − 0.228·5-s + 1.02·6-s + 1.09·7-s − 0.353·8-s + 1.10·9-s + 0.161·10-s − 1.62·11-s − 0.724·12-s + 1.04·13-s − 0.771·14-s + 0.331·15-s + 0.250·16-s − 1.68·17-s − 0.778·18-s − 1.88·19-s − 0.114·20-s − 1.58·21-s + 1.14·22-s − 0.208·23-s + 0.512·24-s − 0.947·25-s − 0.736·26-s − 0.146·27-s + 0.545·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2004542437\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2004542437\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 - T \) |
good | 3 | \( 1 + 2.51T + 3T^{2} \) |
| 5 | \( 1 + 0.511T + 5T^{2} \) |
| 7 | \( 1 - 2.88T + 7T^{2} \) |
| 11 | \( 1 + 5.38T + 11T^{2} \) |
| 13 | \( 1 - 3.75T + 13T^{2} \) |
| 17 | \( 1 + 6.96T + 17T^{2} \) |
| 19 | \( 1 + 8.21T + 19T^{2} \) |
| 29 | \( 1 + 5.85T + 29T^{2} \) |
| 31 | \( 1 + 5.58T + 31T^{2} \) |
| 37 | \( 1 + 3.38T + 37T^{2} \) |
| 41 | \( 1 + 3.72T + 41T^{2} \) |
| 43 | \( 1 - 11.3T + 43T^{2} \) |
| 47 | \( 1 + 7.08T + 47T^{2} \) |
| 53 | \( 1 - 7.83T + 53T^{2} \) |
| 59 | \( 1 - 1.50T + 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 - 14.0T + 67T^{2} \) |
| 71 | \( 1 + 2.01T + 71T^{2} \) |
| 73 | \( 1 - 2.53T + 73T^{2} \) |
| 79 | \( 1 - 1.71T + 79T^{2} \) |
| 83 | \( 1 + 2.22T + 83T^{2} \) |
| 89 | \( 1 - 8.71T + 89T^{2} \) |
| 97 | \( 1 + 9.11T + 97T^{2} \) |
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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
−8.198822504336426156975954777309, −7.37666846386337787839409896648, −6.64174145034245096563746543975, −5.93353737309120518034154907583, −5.37809824686459120223100401886, −4.59142641796243070760917632632, −3.89748081529409276916525283522, −2.32301830036827788977391414071, −1.72864211838059409579463171109, −0.27272816084944141917221126478,
0.27272816084944141917221126478, 1.72864211838059409579463171109, 2.32301830036827788977391414071, 3.89748081529409276916525283522, 4.59142641796243070760917632632, 5.37809824686459120223100401886, 5.93353737309120518034154907583, 6.64174145034245096563746543975, 7.37666846386337787839409896648, 8.198822504336426156975954777309