| L(s) = 1 | − 3-s − 5-s − 4.13·7-s + 9-s + 2.92·11-s + 2.42·13-s + 15-s − 4.15·17-s + 3.84·19-s + 4.13·21-s + 23-s + 25-s − 27-s + 5.64·29-s − 3.22·31-s − 2.92·33-s + 4.13·35-s + 5.55·37-s − 2.42·39-s − 5.06·41-s − 7.41·43-s − 45-s − 3.84·47-s + 10.0·49-s + 4.15·51-s + 8.48·53-s − 2.92·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.56·7-s + 0.333·9-s + 0.883·11-s + 0.672·13-s + 0.258·15-s − 1.00·17-s + 0.881·19-s + 0.901·21-s + 0.208·23-s + 0.200·25-s − 0.192·27-s + 1.04·29-s − 0.578·31-s − 0.509·33-s + 0.698·35-s + 0.912·37-s − 0.388·39-s − 0.790·41-s − 1.13·43-s − 0.149·45-s − 0.560·47-s + 1.43·49-s + 0.581·51-s + 1.16·53-s − 0.395·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 + 4.13T + 7T^{2} \) |
| 11 | \( 1 - 2.92T + 11T^{2} \) |
| 13 | \( 1 - 2.42T + 13T^{2} \) |
| 17 | \( 1 + 4.15T + 17T^{2} \) |
| 19 | \( 1 - 3.84T + 19T^{2} \) |
| 29 | \( 1 - 5.64T + 29T^{2} \) |
| 31 | \( 1 + 3.22T + 31T^{2} \) |
| 37 | \( 1 - 5.55T + 37T^{2} \) |
| 41 | \( 1 + 5.06T + 41T^{2} \) |
| 43 | \( 1 + 7.41T + 43T^{2} \) |
| 47 | \( 1 + 3.84T + 47T^{2} \) |
| 53 | \( 1 - 8.48T + 53T^{2} \) |
| 59 | \( 1 - 3.64T + 59T^{2} \) |
| 61 | \( 1 + 1.51T + 61T^{2} \) |
| 67 | \( 1 + 13.0T + 67T^{2} \) |
| 71 | \( 1 + 7.06T + 71T^{2} \) |
| 73 | \( 1 + 1.57T + 73T^{2} \) |
| 79 | \( 1 - 0.329T + 79T^{2} \) |
| 83 | \( 1 + 13.5T + 83T^{2} \) |
| 89 | \( 1 - 1.08T + 89T^{2} \) |
| 97 | \( 1 - 0.441T + 97T^{2} \) |
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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
−8.640700150307107888421848434344, −7.48345795757012269511501754522, −6.63894232847110506686227555604, −6.41564898954237409595079314646, −5.44008721931475139116523192308, −4.36971010235525083865991362688, −3.64186745344228998441270054467, −2.84250098334476302075560228797, −1.26051612086382796561745648288, 0,
1.26051612086382796561745648288, 2.84250098334476302075560228797, 3.64186745344228998441270054467, 4.36971010235525083865991362688, 5.44008721931475139116523192308, 6.41564898954237409595079314646, 6.63894232847110506686227555604, 7.48345795757012269511501754522, 8.640700150307107888421848434344
