| L(s) = 1 | − 2-s + 0.817·3-s + 4-s − 4.27·5-s − 0.817·6-s − 7-s − 8-s − 2.33·9-s + 4.27·10-s + 1.10·11-s + 0.817·12-s + 3.81·13-s + 14-s − 3.49·15-s + 16-s − 6.42·17-s + 2.33·18-s + 4.12·19-s − 4.27·20-s − 0.817·21-s − 1.10·22-s − 3.07·23-s − 0.817·24-s + 13.2·25-s − 3.81·26-s − 4.35·27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.471·3-s + 0.5·4-s − 1.91·5-s − 0.333·6-s − 0.377·7-s − 0.353·8-s − 0.777·9-s + 1.35·10-s + 0.333·11-s + 0.235·12-s + 1.05·13-s + 0.267·14-s − 0.902·15-s + 0.250·16-s − 1.55·17-s + 0.549·18-s + 0.946·19-s − 0.956·20-s − 0.178·21-s − 0.236·22-s − 0.640·23-s − 0.166·24-s + 2.65·25-s − 0.748·26-s − 0.838·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 431 | \( 1 + T \) |
| good | 3 | \( 1 - 0.817T + 3T^{2} \) |
| 5 | \( 1 + 4.27T + 5T^{2} \) |
| 11 | \( 1 - 1.10T + 11T^{2} \) |
| 13 | \( 1 - 3.81T + 13T^{2} \) |
| 17 | \( 1 + 6.42T + 17T^{2} \) |
| 19 | \( 1 - 4.12T + 19T^{2} \) |
| 23 | \( 1 + 3.07T + 23T^{2} \) |
| 29 | \( 1 - 5.16T + 29T^{2} \) |
| 31 | \( 1 + 6.45T + 31T^{2} \) |
| 37 | \( 1 - 7.19T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 - 2.73T + 43T^{2} \) |
| 47 | \( 1 - 8.58T + 47T^{2} \) |
| 53 | \( 1 + 1.42T + 53T^{2} \) |
| 59 | \( 1 + 12.7T + 59T^{2} \) |
| 61 | \( 1 - 7.89T + 61T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 - 0.876T + 71T^{2} \) |
| 73 | \( 1 + 2.98T + 73T^{2} \) |
| 79 | \( 1 + 7.83T + 79T^{2} \) |
| 83 | \( 1 - 5.84T + 83T^{2} \) |
| 89 | \( 1 - 7.08T + 89T^{2} \) |
| 97 | \( 1 - 11.6T + 97T^{2} \) |
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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
−7.74550470997199856608091733539, −7.38159189259142685398321854373, −6.47237541278533140182540138770, −5.82873670715515226803548717577, −4.49804646075724507940712973806, −3.91344878079056646015566795152, −3.21286186732496548374110769298, −2.47341546972695929493130070516, −0.989946311367132953764585493067, 0,
0.989946311367132953764585493067, 2.47341546972695929493130070516, 3.21286186732496548374110769298, 3.91344878079056646015566795152, 4.49804646075724507940712973806, 5.82873670715515226803548717577, 6.47237541278533140182540138770, 7.38159189259142685398321854373, 7.74550470997199856608091733539
