L(s) = 1 | − 5-s + 3.20·7-s − 5.20·11-s + 5.45·13-s − 4·17-s + 19-s + 8.65·23-s + 25-s + 3.20·29-s − 2·31-s − 3.20·35-s − 11.8·37-s + 9.85·41-s − 3.45·43-s − 8.65·47-s + 3.25·49-s + 10.6·53-s + 5.20·55-s + 13.0·59-s + 3.74·61-s − 5.45·65-s + 4·67-s − 6.90·71-s + 6·73-s − 16.6·77-s − 6.15·79-s + 8.40·83-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 1.21·7-s − 1.56·11-s + 1.51·13-s − 0.970·17-s + 0.229·19-s + 1.80·23-s + 0.200·25-s + 0.594·29-s − 0.359·31-s − 0.541·35-s − 1.94·37-s + 1.53·41-s − 0.526·43-s − 1.26·47-s + 0.464·49-s + 1.46·53-s + 0.701·55-s + 1.70·59-s + 0.479·61-s − 0.676·65-s + 0.488·67-s − 0.819·71-s + 0.702·73-s − 1.89·77-s − 0.692·79-s + 0.922·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.041966435\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.041966435\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 - T \) |
good | 7 | \( 1 - 3.20T + 7T^{2} \) |
| 11 | \( 1 + 5.20T + 11T^{2} \) |
| 13 | \( 1 - 5.45T + 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 23 | \( 1 - 8.65T + 23T^{2} \) |
| 29 | \( 1 - 3.20T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + 11.8T + 37T^{2} \) |
| 41 | \( 1 - 9.85T + 41T^{2} \) |
| 43 | \( 1 + 3.45T + 43T^{2} \) |
| 47 | \( 1 + 8.65T + 47T^{2} \) |
| 53 | \( 1 - 10.6T + 53T^{2} \) |
| 59 | \( 1 - 13.0T + 59T^{2} \) |
| 61 | \( 1 - 3.74T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 6.90T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 6.15T + 79T^{2} \) |
| 83 | \( 1 - 8.40T + 83T^{2} \) |
| 89 | \( 1 + 13.8T + 89T^{2} \) |
| 97 | \( 1 - 0.142T + 97T^{2} \) |
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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
−8.188199105191009634253845765490, −7.24610401641665183509485782090, −6.78150661206004317132619650485, −5.62348746928534938394869660433, −5.13611453727594716769207104007, −4.48819527642475406629501672974, −3.58284454442487163696595953688, −2.74324356271774964766129324211, −1.78396171876200716098222588106, −0.74834249765321032478226912845,
0.74834249765321032478226912845, 1.78396171876200716098222588106, 2.74324356271774964766129324211, 3.58284454442487163696595953688, 4.48819527642475406629501672974, 5.13611453727594716769207104007, 5.62348746928534938394869660433, 6.78150661206004317132619650485, 7.24610401641665183509485782090, 8.188199105191009634253845765490