L(s) = 1 | − 3-s − 0.580·5-s + 3.10·7-s + 9-s + 0.969·11-s + 5.69·13-s + 0.580·15-s − 3.05·17-s − 4.37·19-s − 3.10·21-s + 0.0518·23-s − 4.66·25-s − 27-s − 1.78·29-s − 2.04·31-s − 0.969·33-s − 1.80·35-s − 7.32·37-s − 5.69·39-s − 7.34·41-s − 6.53·43-s − 0.580·45-s − 2.56·47-s + 2.65·49-s + 3.05·51-s − 8.86·53-s − 0.562·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.259·5-s + 1.17·7-s + 0.333·9-s + 0.292·11-s + 1.57·13-s + 0.149·15-s − 0.740·17-s − 1.00·19-s − 0.678·21-s + 0.0108·23-s − 0.932·25-s − 0.192·27-s − 0.332·29-s − 0.367·31-s − 0.168·33-s − 0.304·35-s − 1.20·37-s − 0.912·39-s − 1.14·41-s − 0.996·43-s − 0.0865·45-s − 0.373·47-s + 0.379·49-s + 0.427·51-s − 1.21·53-s − 0.0759·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 251 | \( 1 + T \) |
good | 5 | \( 1 + 0.580T + 5T^{2} \) |
| 7 | \( 1 - 3.10T + 7T^{2} \) |
| 11 | \( 1 - 0.969T + 11T^{2} \) |
| 13 | \( 1 - 5.69T + 13T^{2} \) |
| 17 | \( 1 + 3.05T + 17T^{2} \) |
| 19 | \( 1 + 4.37T + 19T^{2} \) |
| 23 | \( 1 - 0.0518T + 23T^{2} \) |
| 29 | \( 1 + 1.78T + 29T^{2} \) |
| 31 | \( 1 + 2.04T + 31T^{2} \) |
| 37 | \( 1 + 7.32T + 37T^{2} \) |
| 41 | \( 1 + 7.34T + 41T^{2} \) |
| 43 | \( 1 + 6.53T + 43T^{2} \) |
| 47 | \( 1 + 2.56T + 47T^{2} \) |
| 53 | \( 1 + 8.86T + 53T^{2} \) |
| 59 | \( 1 + 3.94T + 59T^{2} \) |
| 61 | \( 1 - 6.37T + 61T^{2} \) |
| 67 | \( 1 - 1.87T + 67T^{2} \) |
| 71 | \( 1 + 0.104T + 71T^{2} \) |
| 73 | \( 1 - 14.9T + 73T^{2} \) |
| 79 | \( 1 + 12.0T + 79T^{2} \) |
| 83 | \( 1 - 12.8T + 83T^{2} \) |
| 89 | \( 1 - 0.546T + 89T^{2} \) |
| 97 | \( 1 + 19.4T + 97T^{2} \) |
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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
−7.897340102790892215019782791244, −6.80967757232926575123326339622, −6.41011712706769219401536085325, −5.52503813471135074948224632330, −4.86083095002445230216090991586, −4.08023407440409418503389309389, −3.49384408431309014878213530821, −1.97504270527231337636540197043, −1.43844745439108782308009489668, 0,
1.43844745439108782308009489668, 1.97504270527231337636540197043, 3.49384408431309014878213530821, 4.08023407440409418503389309389, 4.86083095002445230216090991586, 5.52503813471135074948224632330, 6.41011712706769219401536085325, 6.80967757232926575123326339622, 7.897340102790892215019782791244