| L(s) = 1 | − 2.72·2-s + 5.44·4-s − 0.363·7-s − 9.40·8-s − 5.69·11-s − 2.16·13-s + 0.992·14-s + 14.7·16-s − 1.91·17-s + 1.10·19-s + 15.5·22-s − 7.73·23-s + 5.90·26-s − 1.98·28-s + 6.52·29-s − 31-s − 21.4·32-s + 5.22·34-s + 5.51·37-s − 3.01·38-s + 8.62·41-s − 9.99·43-s − 31.0·44-s + 21.1·46-s + 4.18·47-s − 6.86·49-s − 11.7·52-s + ⋯ |
| L(s) = 1 | − 1.92·2-s + 2.72·4-s − 0.137·7-s − 3.32·8-s − 1.71·11-s − 0.600·13-s + 0.265·14-s + 3.69·16-s − 0.464·17-s + 0.253·19-s + 3.31·22-s − 1.61·23-s + 1.15·26-s − 0.374·28-s + 1.21·29-s − 0.179·31-s − 3.79·32-s + 0.896·34-s + 0.905·37-s − 0.489·38-s + 1.34·41-s − 1.52·43-s − 4.67·44-s + 3.11·46-s + 0.610·47-s − 0.981·49-s − 1.63·52-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
| \(L(1)\) |
\(\approx\) |
\(0.2843051332\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2843051332\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.72T + 2T^{2} \) |
| 7 | \( 1 + 0.363T + 7T^{2} \) |
| 11 | \( 1 + 5.69T + 11T^{2} \) |
| 13 | \( 1 + 2.16T + 13T^{2} \) |
| 17 | \( 1 + 1.91T + 17T^{2} \) |
| 19 | \( 1 - 1.10T + 19T^{2} \) |
| 23 | \( 1 + 7.73T + 23T^{2} \) |
| 29 | \( 1 - 6.52T + 29T^{2} \) |
| 37 | \( 1 - 5.51T + 37T^{2} \) |
| 41 | \( 1 - 8.62T + 41T^{2} \) |
| 43 | \( 1 + 9.99T + 43T^{2} \) |
| 47 | \( 1 - 4.18T + 47T^{2} \) |
| 53 | \( 1 - 5.03T + 53T^{2} \) |
| 59 | \( 1 + 7.34T + 59T^{2} \) |
| 61 | \( 1 - 1.24T + 61T^{2} \) |
| 67 | \( 1 - 1.06T + 67T^{2} \) |
| 71 | \( 1 + 11.7T + 71T^{2} \) |
| 73 | \( 1 + 11.7T + 73T^{2} \) |
| 79 | \( 1 - 0.773T + 79T^{2} \) |
| 83 | \( 1 + 8.37T + 83T^{2} \) |
| 89 | \( 1 - 5.48T + 89T^{2} \) |
| 97 | \( 1 + 7.34T + 97T^{2} \) |
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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
−8.022916289204071574420306602444, −7.58031317041270199022700195328, −6.86825122272724505952479168879, −6.10367586672595098078689808066, −5.44852420020653514369030511990, −4.35220022102355939439531619970, −2.92209429433729519155534117324, −2.55040606455613686520567430532, −1.62535068259826333862389073291, −0.35047261926425054290843369839,
0.35047261926425054290843369839, 1.62535068259826333862389073291, 2.55040606455613686520567430532, 2.92209429433729519155534117324, 4.35220022102355939439531619970, 5.44852420020653514369030511990, 6.10367586672595098078689808066, 6.86825122272724505952479168879, 7.58031317041270199022700195328, 8.022916289204071574420306602444
