L(s) = 1 | − 1.73·3-s + 5-s − 0.267·7-s − 3.46·13-s − 1.73·15-s − 2·19-s + 0.464·21-s + 7.46·23-s + 25-s + 5.19·27-s + 4.92·29-s − 1.46·31-s − 0.267·35-s − 4·37-s + 5.99·39-s + 1.92·41-s − 1.73·43-s + 6.66·47-s − 6.92·49-s − 7.46·53-s + 3.46·57-s − 1.46·59-s − 1.53·61-s − 3.46·65-s + 5.73·67-s − 12.9·69-s + ⋯ |
L(s) = 1 | − 1.00·3-s + 0.447·5-s − 0.101·7-s − 0.960·13-s − 0.447·15-s − 0.458·19-s + 0.101·21-s + 1.55·23-s + 0.200·25-s + 1.00·27-s + 0.915·29-s − 0.262·31-s − 0.0452·35-s − 0.657·37-s + 0.960·39-s + 0.301·41-s − 0.264·43-s + 0.971·47-s − 0.989·49-s − 1.02·53-s + 0.458·57-s − 0.190·59-s − 0.196·61-s − 0.429·65-s + 0.700·67-s − 1.55·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 3 | \( 1 + 1.73T + 3T^{2} \) |
| 7 | \( 1 + 0.267T + 7T^{2} \) |
| 13 | \( 1 + 3.46T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 7.46T + 23T^{2} \) |
| 29 | \( 1 - 4.92T + 29T^{2} \) |
| 31 | \( 1 + 1.46T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 - 1.92T + 41T^{2} \) |
| 43 | \( 1 + 1.73T + 43T^{2} \) |
| 47 | \( 1 - 6.66T + 47T^{2} \) |
| 53 | \( 1 + 7.46T + 53T^{2} \) |
| 59 | \( 1 + 1.46T + 59T^{2} \) |
| 61 | \( 1 + 1.53T + 61T^{2} \) |
| 67 | \( 1 - 5.73T + 67T^{2} \) |
| 71 | \( 1 - 2.53T + 71T^{2} \) |
| 73 | \( 1 + 1.07T + 73T^{2} \) |
| 79 | \( 1 + 14.3T + 79T^{2} \) |
| 83 | \( 1 - 7.46T + 83T^{2} \) |
| 89 | \( 1 - 15.9T + 89T^{2} \) |
| 97 | \( 1 + 14.3T + 97T^{2} \) |
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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
−7.84753234068020269232166021703, −6.89612588365918559484952764328, −6.53709702176599717484256472948, −5.64039860097785044624081189762, −5.07257763953046251388323293577, −4.48393011747447579454550068455, −3.20209625287725742123447739894, −2.42808760330202337110595724028, −1.18838819117394696200069298759, 0,
1.18838819117394696200069298759, 2.42808760330202337110595724028, 3.20209625287725742123447739894, 4.48393011747447579454550068455, 5.07257763953046251388323293577, 5.64039860097785044624081189762, 6.53709702176599717484256472948, 6.89612588365918559484952764328, 7.84753234068020269232166021703