L(s) = 1 | + 2-s − 2.27·3-s + 4-s − 1.81·5-s − 2.27·6-s + 2.75·7-s + 8-s + 2.15·9-s − 1.81·10-s − 3.69·11-s − 2.27·12-s + 4.65·13-s + 2.75·14-s + 4.11·15-s + 16-s + 4.00·17-s + 2.15·18-s + 6.43·19-s − 1.81·20-s − 6.25·21-s − 3.69·22-s + 23-s − 2.27·24-s − 1.72·25-s + 4.65·26-s + 1.92·27-s + 2.75·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.31·3-s + 0.5·4-s − 0.809·5-s − 0.926·6-s + 1.04·7-s + 0.353·8-s + 0.717·9-s − 0.572·10-s − 1.11·11-s − 0.655·12-s + 1.29·13-s + 0.736·14-s + 1.06·15-s + 0.250·16-s + 0.972·17-s + 0.507·18-s + 1.47·19-s − 0.404·20-s − 1.36·21-s − 0.788·22-s + 0.208·23-s − 0.463·24-s − 0.344·25-s + 0.913·26-s + 0.369·27-s + 0.520·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.909549599\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.909549599\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 - T \) |
good | 3 | \( 1 + 2.27T + 3T^{2} \) |
| 5 | \( 1 + 1.81T + 5T^{2} \) |
| 7 | \( 1 - 2.75T + 7T^{2} \) |
| 11 | \( 1 + 3.69T + 11T^{2} \) |
| 13 | \( 1 - 4.65T + 13T^{2} \) |
| 17 | \( 1 - 4.00T + 17T^{2} \) |
| 19 | \( 1 - 6.43T + 19T^{2} \) |
| 29 | \( 1 + 0.101T + 29T^{2} \) |
| 31 | \( 1 - 2.69T + 31T^{2} \) |
| 37 | \( 1 + 1.93T + 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 - 1.32T + 43T^{2} \) |
| 47 | \( 1 + 7.72T + 47T^{2} \) |
| 53 | \( 1 + 5.32T + 53T^{2} \) |
| 59 | \( 1 + 2.92T + 59T^{2} \) |
| 61 | \( 1 - 12.7T + 61T^{2} \) |
| 67 | \( 1 - 4.34T + 67T^{2} \) |
| 71 | \( 1 + 7.06T + 71T^{2} \) |
| 73 | \( 1 - 8.48T + 73T^{2} \) |
| 79 | \( 1 - 15.8T + 79T^{2} \) |
| 83 | \( 1 - 1.11T + 83T^{2} \) |
| 89 | \( 1 + 2.13T + 89T^{2} \) |
| 97 | \( 1 - 13.2T + 97T^{2} \) |
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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
−8.074163471574661310793910557448, −7.28987464439337775741034910818, −6.49649585176933529821749791779, −5.68079874945436791050519542183, −5.17513005483254586355653845022, −4.78579879050721401626286901732, −3.71147890786747784486881828080, −3.10594636116245991843614502002, −1.66001656639752492472134034170, −0.73365921414613626995971755015,
0.73365921414613626995971755015, 1.66001656639752492472134034170, 3.10594636116245991843614502002, 3.71147890786747784486881828080, 4.78579879050721401626286901732, 5.17513005483254586355653845022, 5.68079874945436791050519542183, 6.49649585176933529821749791779, 7.28987464439337775741034910818, 8.074163471574661310793910557448