| L(s) = 1 | − 0.801·2-s − 1.35·4-s − 5-s − 3.24·7-s + 2.69·8-s + 0.801·10-s − 2.49·11-s − 1.55·13-s + 2.60·14-s + 0.554·16-s + 6.29·17-s − 4.89·19-s + 1.35·20-s + 2·22-s + 1.10·23-s + 25-s + 1.24·26-s + 4.40·28-s + 9.40·29-s + 0.713·31-s − 5.82·32-s − 5.04·34-s + 3.24·35-s + 6.29·37-s + 3.92·38-s − 2.69·40-s + 4.53·41-s + ⋯ |
| L(s) = 1 | − 0.567·2-s − 0.678·4-s − 0.447·5-s − 1.22·7-s + 0.951·8-s + 0.253·10-s − 0.751·11-s − 0.431·13-s + 0.695·14-s + 0.138·16-s + 1.52·17-s − 1.12·19-s + 0.303·20-s + 0.426·22-s + 0.231·23-s + 0.200·25-s + 0.244·26-s + 0.832·28-s + 1.74·29-s + 0.128·31-s − 1.03·32-s − 0.865·34-s + 0.548·35-s + 1.03·37-s + 0.636·38-s − 0.425·40-s + 0.707·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 89 | \( 1 - T \) |
| good | 2 | \( 1 + 0.801T + 2T^{2} \) |
| 7 | \( 1 + 3.24T + 7T^{2} \) |
| 11 | \( 1 + 2.49T + 11T^{2} \) |
| 13 | \( 1 + 1.55T + 13T^{2} \) |
| 17 | \( 1 - 6.29T + 17T^{2} \) |
| 19 | \( 1 + 4.89T + 19T^{2} \) |
| 23 | \( 1 - 1.10T + 23T^{2} \) |
| 29 | \( 1 - 9.40T + 29T^{2} \) |
| 31 | \( 1 - 0.713T + 31T^{2} \) |
| 37 | \( 1 - 6.29T + 37T^{2} \) |
| 41 | \( 1 - 4.53T + 41T^{2} \) |
| 43 | \( 1 + 2.19T + 43T^{2} \) |
| 47 | \( 1 + 2.53T + 47T^{2} \) |
| 53 | \( 1 + 3.56T + 53T^{2} \) |
| 59 | \( 1 - 14.2T + 59T^{2} \) |
| 61 | \( 1 + 8.49T + 61T^{2} \) |
| 67 | \( 1 + 4.09T + 67T^{2} \) |
| 71 | \( 1 - 9.08T + 71T^{2} \) |
| 73 | \( 1 + 1.72T + 73T^{2} \) |
| 79 | \( 1 + 2.79T + 79T^{2} \) |
| 83 | \( 1 + 17.7T + 83T^{2} \) |
| 97 | \( 1 - 13.9T + 97T^{2} \) |
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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
−8.118850971261798605219362785538, −7.58251871218208636548757950359, −6.72216429381638992258726676342, −5.91228167203727121124133962170, −5.00191437530549149505854175546, −4.27709725495812923029753484712, −3.36249613406631931611864051014, −2.59484714148954947059025313772, −1.00143698708267362818515375017, 0,
1.00143698708267362818515375017, 2.59484714148954947059025313772, 3.36249613406631931611864051014, 4.27709725495812923029753484712, 5.00191437530549149505854175546, 5.91228167203727121124133962170, 6.72216429381638992258726676342, 7.58251871218208636548757950359, 8.118850971261798605219362785538
