| L(s) = 1 | + 2·5-s + 7-s + 4·11-s − 6·13-s + 4·17-s − 4·19-s − 25-s + 10·29-s − 2·31-s + 2·35-s − 37-s + 2·41-s + 4·43-s + 49-s − 6·53-s + 8·55-s − 14·61-s − 12·65-s − 12·67-s − 8·71-s − 14·73-s + 4·77-s − 12·79-s + 6·83-s + 8·85-s − 6·91-s − 8·95-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.377·7-s + 1.20·11-s − 1.66·13-s + 0.970·17-s − 0.917·19-s − 1/5·25-s + 1.85·29-s − 0.359·31-s + 0.338·35-s − 0.164·37-s + 0.312·41-s + 0.609·43-s + 1/7·49-s − 0.824·53-s + 1.07·55-s − 1.79·61-s − 1.48·65-s − 1.46·67-s − 0.949·71-s − 1.63·73-s + 0.455·77-s − 1.35·79-s + 0.658·83-s + 0.867·85-s − 0.628·91-s − 0.820·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 37 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−14.91315576787768, −14.53467231696006, −14.14044523209417, −13.80279468987255, −12.95817993733923, −12.48019221092874, −11.95685103044077, −11.69894998048986, −10.78802390008739, −10.13317821909504, −10.04876461293268, −9.107355724774731, −9.033202054028157, −8.115147293835445, −7.522125019317980, −7.045371318205750, −6.202847561305539, −6.016999358721570, −5.148005221717656, −4.587786985187470, −4.125919335898435, −3.074079616941864, −2.570687838082286, −1.733734059040485, −1.226027886664284, 0,
1.226027886664284, 1.733734059040485, 2.570687838082286, 3.074079616941864, 4.125919335898435, 4.587786985187470, 5.148005221717656, 6.016999358721570, 6.202847561305539, 7.045371318205750, 7.522125019317980, 8.115147293835445, 9.033202054028157, 9.107355724774731, 10.04876461293268, 10.13317821909504, 10.78802390008739, 11.69894998048986, 11.95685103044077, 12.48019221092874, 12.95817993733923, 13.80279468987255, 14.14044523209417, 14.53467231696006, 14.91315576787768
