| L(s) = 1 | + 0.667·2-s − 1.55·4-s + 3.64·7-s − 2.37·8-s + 1.15·11-s + 5.71·13-s + 2.43·14-s + 1.52·16-s + 1.37·17-s + 2.28·19-s + 0.772·22-s − 5.17·23-s + 3.81·26-s − 5.66·28-s + 5.51·29-s − 31-s + 5.76·32-s + 0.920·34-s + 2.77·37-s + 1.52·38-s + 0.122·41-s + 1.11·43-s − 1.79·44-s − 3.45·46-s + 8.92·47-s + 6.26·49-s − 8.87·52-s + ⋯ |
| L(s) = 1 | + 0.471·2-s − 0.777·4-s + 1.37·7-s − 0.838·8-s + 0.348·11-s + 1.58·13-s + 0.649·14-s + 0.381·16-s + 0.334·17-s + 0.524·19-s + 0.164·22-s − 1.07·23-s + 0.747·26-s − 1.07·28-s + 1.02·29-s − 0.179·31-s + 1.01·32-s + 0.157·34-s + 0.455·37-s + 0.247·38-s + 0.0191·41-s + 0.170·43-s − 0.271·44-s − 0.508·46-s + 1.30·47-s + 0.895·49-s − 1.23·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.858708218\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.858708218\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 - 0.667T + 2T^{2} \) |
| 7 | \( 1 - 3.64T + 7T^{2} \) |
| 11 | \( 1 - 1.15T + 11T^{2} \) |
| 13 | \( 1 - 5.71T + 13T^{2} \) |
| 17 | \( 1 - 1.37T + 17T^{2} \) |
| 19 | \( 1 - 2.28T + 19T^{2} \) |
| 23 | \( 1 + 5.17T + 23T^{2} \) |
| 29 | \( 1 - 5.51T + 29T^{2} \) |
| 37 | \( 1 - 2.77T + 37T^{2} \) |
| 41 | \( 1 - 0.122T + 41T^{2} \) |
| 43 | \( 1 - 1.11T + 43T^{2} \) |
| 47 | \( 1 - 8.92T + 47T^{2} \) |
| 53 | \( 1 + 3.42T + 53T^{2} \) |
| 59 | \( 1 + 0.542T + 59T^{2} \) |
| 61 | \( 1 - 10.4T + 61T^{2} \) |
| 67 | \( 1 + 15.8T + 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 + 8.34T + 73T^{2} \) |
| 79 | \( 1 - 5.83T + 79T^{2} \) |
| 83 | \( 1 + 16.4T + 83T^{2} \) |
| 89 | \( 1 - 8.27T + 89T^{2} \) |
| 97 | \( 1 - 9.97T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.016369365649395529547974595010, −7.42494756165249304115946397924, −6.20026539580077791985372868469, −5.83693440459044320994649684166, −5.02438711257654474761618272315, −4.32559887843461929294438487361, −3.83096025121202409342034075224, −2.92156368686172657300740578699, −1.65400963698065959987354180084, −0.885623569134791076306177710215,
0.885623569134791076306177710215, 1.65400963698065959987354180084, 2.92156368686172657300740578699, 3.83096025121202409342034075224, 4.32559887843461929294438487361, 5.02438711257654474761618272315, 5.83693440459044320994649684166, 6.20026539580077791985372868469, 7.42494756165249304115946397924, 8.016369365649395529547974595010
