L(s) = 1 | + 5-s − 2·7-s − 11-s + 8·17-s − 8·19-s − 4·23-s + 25-s + 6·29-s − 2·35-s + 6·37-s + 2·41-s + 2·43-s + 4·47-s − 3·49-s + 2·53-s − 55-s + 12·59-s − 6·61-s + 8·67-s − 8·73-s + 2·77-s + 4·79-s + 6·83-s + 8·85-s + 10·89-s − 8·95-s − 10·97-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.755·7-s − 0.301·11-s + 1.94·17-s − 1.83·19-s − 0.834·23-s + 1/5·25-s + 1.11·29-s − 0.338·35-s + 0.986·37-s + 0.312·41-s + 0.304·43-s + 0.583·47-s − 3/7·49-s + 0.274·53-s − 0.134·55-s + 1.56·59-s − 0.768·61-s + 0.977·67-s − 0.936·73-s + 0.227·77-s + 0.450·79-s + 0.658·83-s + 0.867·85-s + 1.05·89-s − 0.820·95-s − 1.01·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.757932927\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.757932927\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.378529573197428734409008106681, −7.83877309088964992232876588117, −6.89297214817942897916236648043, −6.14340315396523346146302421076, −5.70325274306668947557808369172, −4.67076738944322484715966029500, −3.80817728324857729378799634204, −2.92165762439534383333189009412, −2.07609416527182384876181592530, −0.75347698742480715400167322922,
0.75347698742480715400167322922, 2.07609416527182384876181592530, 2.92165762439534383333189009412, 3.80817728324857729378799634204, 4.67076738944322484715966029500, 5.70325274306668947557808369172, 6.14340315396523346146302421076, 6.89297214817942897916236648043, 7.83877309088964992232876588117, 8.378529573197428734409008106681