| L(s) = 1 | − 1.21·2-s + 3-s − 0.525·4-s − 1.21·6-s − 2.59·7-s + 3.06·8-s + 9-s − 6.11·11-s − 0.525·12-s − 13-s + 3.14·14-s − 2.67·16-s + 4.37·17-s − 1.21·18-s + 4.14·19-s − 2.59·21-s + 7.42·22-s + 7.95·23-s + 3.06·24-s + 1.21·26-s + 27-s + 1.36·28-s − 3·29-s + 5.36·31-s − 2.88·32-s − 6.11·33-s − 5.31·34-s + ⋯ |
| L(s) = 1 | − 0.858·2-s + 0.577·3-s − 0.262·4-s − 0.495·6-s − 0.979·7-s + 1.08·8-s + 0.333·9-s − 1.84·11-s − 0.151·12-s − 0.277·13-s + 0.841·14-s − 0.668·16-s + 1.06·17-s − 0.286·18-s + 0.951·19-s − 0.565·21-s + 1.58·22-s + 1.65·23-s + 0.625·24-s + 0.238·26-s + 0.192·27-s + 0.257·28-s − 0.557·29-s + 0.963·31-s − 0.510·32-s − 1.06·33-s − 0.911·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8671053844\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8671053844\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + 1.21T + 2T^{2} \) |
| 7 | \( 1 + 2.59T + 7T^{2} \) |
| 11 | \( 1 + 6.11T + 11T^{2} \) |
| 17 | \( 1 - 4.37T + 17T^{2} \) |
| 19 | \( 1 - 4.14T + 19T^{2} \) |
| 23 | \( 1 - 7.95T + 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 - 5.36T + 31T^{2} \) |
| 37 | \( 1 - 6.90T + 37T^{2} \) |
| 41 | \( 1 + 9.19T + 41T^{2} \) |
| 43 | \( 1 - 11.1T + 43T^{2} \) |
| 47 | \( 1 + 1.21T + 47T^{2} \) |
| 53 | \( 1 - 4.95T + 53T^{2} \) |
| 59 | \( 1 - 5.44T + 59T^{2} \) |
| 61 | \( 1 - 9.99T + 61T^{2} \) |
| 67 | \( 1 + 4.87T + 67T^{2} \) |
| 71 | \( 1 - 1.39T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 + 12.3T + 79T^{2} \) |
| 83 | \( 1 + 13.2T + 83T^{2} \) |
| 89 | \( 1 - 8.04T + 89T^{2} \) |
| 97 | \( 1 - 15.3T + 97T^{2} \) |
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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
−9.944077544407112764315997161787, −9.268329377697019859748151080758, −8.374495904931836180641793617272, −7.64953480711241628913584740576, −7.08887653934800736059656474212, −5.59441979318281416250220962454, −4.81063731372148289637626048963, −3.40621237587882170457010554964, −2.58253296824175436567424405722, −0.806120746061531410740940150160,
0.806120746061531410740940150160, 2.58253296824175436567424405722, 3.40621237587882170457010554964, 4.81063731372148289637626048963, 5.59441979318281416250220962454, 7.08887653934800736059656474212, 7.64953480711241628913584740576, 8.374495904931836180641793617272, 9.268329377697019859748151080758, 9.944077544407112764315997161787
