L(s) = 1 | − 0.541·2-s − 1.70·4-s − 1.08·7-s + 2.00·8-s − 2.91·11-s − 0.966·13-s + 0.589·14-s + 2.32·16-s + 2.65·17-s − 1.96·19-s + 1.57·22-s − 6.83·23-s + 0.523·26-s + 1.85·28-s + 6.81·29-s + 2.48·31-s − 5.27·32-s − 1.44·34-s + 8.22·37-s + 1.06·38-s + 10.5·41-s + 7.72·43-s + 4.97·44-s + 3.70·46-s + 9.02·47-s − 5.81·49-s + 1.64·52-s + ⋯ |
L(s) = 1 | − 0.383·2-s − 0.853·4-s − 0.411·7-s + 0.710·8-s − 0.878·11-s − 0.267·13-s + 0.157·14-s + 0.581·16-s + 0.645·17-s − 0.451·19-s + 0.336·22-s − 1.42·23-s + 0.102·26-s + 0.350·28-s + 1.26·29-s + 0.446·31-s − 0.932·32-s − 0.247·34-s + 1.35·37-s + 0.172·38-s + 1.65·41-s + 1.17·43-s + 0.749·44-s + 0.546·46-s + 1.31·47-s − 0.830·49-s + 0.228·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + 0.541T + 2T^{2} \) |
| 7 | \( 1 + 1.08T + 7T^{2} \) |
| 11 | \( 1 + 2.91T + 11T^{2} \) |
| 13 | \( 1 + 0.966T + 13T^{2} \) |
| 17 | \( 1 - 2.65T + 17T^{2} \) |
| 19 | \( 1 + 1.96T + 19T^{2} \) |
| 23 | \( 1 + 6.83T + 23T^{2} \) |
| 29 | \( 1 - 6.81T + 29T^{2} \) |
| 31 | \( 1 - 2.48T + 31T^{2} \) |
| 37 | \( 1 - 8.22T + 37T^{2} \) |
| 41 | \( 1 - 10.5T + 41T^{2} \) |
| 43 | \( 1 - 7.72T + 43T^{2} \) |
| 47 | \( 1 - 9.02T + 47T^{2} \) |
| 53 | \( 1 + 5.53T + 53T^{2} \) |
| 59 | \( 1 + 13.2T + 59T^{2} \) |
| 61 | \( 1 - 3.64T + 61T^{2} \) |
| 67 | \( 1 + 8.84T + 67T^{2} \) |
| 71 | \( 1 + 7.81T + 71T^{2} \) |
| 73 | \( 1 + 15.6T + 73T^{2} \) |
| 79 | \( 1 + 6.60T + 79T^{2} \) |
| 83 | \( 1 - 1.89T + 83T^{2} \) |
| 89 | \( 1 + 1.14T + 89T^{2} \) |
| 97 | \( 1 - 6.19T + 97T^{2} \) |
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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
−7.72520836940492320999935168519, −7.49144611731552239007876828320, −6.07372258437220583886372470031, −5.85226187201739563214816363794, −4.60368601746362948663273744766, −4.35061276794740853613902616287, −3.19307716242613237605822459817, −2.39106372204962878119590993892, −1.06552298075319220856487162639, 0,
1.06552298075319220856487162639, 2.39106372204962878119590993892, 3.19307716242613237605822459817, 4.35061276794740853613902616287, 4.60368601746362948663273744766, 5.85226187201739563214816363794, 6.07372258437220583886372470031, 7.49144611731552239007876828320, 7.72520836940492320999935168519