| L(s) = 1 | − 2.45·2-s + 3-s + 4.01·4-s − 0.0682·5-s − 2.45·6-s − 7-s − 4.94·8-s + 9-s + 0.167·10-s + 0.678·11-s + 4.01·12-s + 2.45·14-s − 0.0682·15-s + 4.10·16-s − 0.934·17-s − 2.45·18-s − 2.06·19-s − 0.274·20-s − 21-s − 1.66·22-s + 4.29·23-s − 4.94·24-s − 4.99·25-s + 27-s − 4.01·28-s + 8.01·29-s + 0.167·30-s + ⋯ |
| L(s) = 1 | − 1.73·2-s + 0.577·3-s + 2.00·4-s − 0.0305·5-s − 1.00·6-s − 0.377·7-s − 1.74·8-s + 0.333·9-s + 0.0529·10-s + 0.204·11-s + 1.15·12-s + 0.655·14-s − 0.0176·15-s + 1.02·16-s − 0.226·17-s − 0.578·18-s − 0.474·19-s − 0.0613·20-s − 0.218·21-s − 0.354·22-s + 0.896·23-s − 1.01·24-s − 0.999·25-s + 0.192·27-s − 0.759·28-s + 1.48·29-s + 0.0305·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.45T + 2T^{2} \) |
| 5 | \( 1 + 0.0682T + 5T^{2} \) |
| 11 | \( 1 - 0.678T + 11T^{2} \) |
| 17 | \( 1 + 0.934T + 17T^{2} \) |
| 19 | \( 1 + 2.06T + 19T^{2} \) |
| 23 | \( 1 - 4.29T + 23T^{2} \) |
| 29 | \( 1 - 8.01T + 29T^{2} \) |
| 31 | \( 1 + 10.0T + 31T^{2} \) |
| 37 | \( 1 - 0.527T + 37T^{2} \) |
| 41 | \( 1 + 5.13T + 41T^{2} \) |
| 43 | \( 1 + 1.78T + 43T^{2} \) |
| 47 | \( 1 + 4.22T + 47T^{2} \) |
| 53 | \( 1 + 3.21T + 53T^{2} \) |
| 59 | \( 1 - 8.32T + 59T^{2} \) |
| 61 | \( 1 - 1.01T + 61T^{2} \) |
| 67 | \( 1 - 7.30T + 67T^{2} \) |
| 71 | \( 1 + 9.91T + 71T^{2} \) |
| 73 | \( 1 - 3.42T + 73T^{2} \) |
| 79 | \( 1 + 14.0T + 79T^{2} \) |
| 83 | \( 1 - 7.03T + 83T^{2} \) |
| 89 | \( 1 - 15.7T + 89T^{2} \) |
| 97 | \( 1 + 16.6T + 97T^{2} \) |
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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
−8.330986158510072524797455481883, −7.72988561029893290499019987941, −6.88606092898951272818680059876, −6.50944796095509936130898755815, −5.32551941583404770204373869833, −4.10465020782197851481324317619, −3.10671092987798521382663998396, −2.22058275560937481563322001178, −1.32168063045229017690289531950, 0,
1.32168063045229017690289531950, 2.22058275560937481563322001178, 3.10671092987798521382663998396, 4.10465020782197851481324317619, 5.32551941583404770204373869833, 6.50944796095509936130898755815, 6.88606092898951272818680059876, 7.72988561029893290499019987941, 8.330986158510072524797455481883
