| L(s) = 1 | + 5-s − 3.44·7-s + 1.44·11-s − 5.89·13-s + 4.89·17-s + 4·19-s + 5.44·23-s − 4·25-s − 0.101·29-s − 2.55·31-s − 3.44·35-s + 0.898·37-s − 11.8·41-s − 2.34·43-s + 6.34·47-s + 4.89·49-s − 8.89·53-s + 1.44·55-s + 14.3·59-s + 7.89·61-s − 5.89·65-s + 12.3·67-s − 7.79·71-s − 4.89·73-s − 5·77-s − 13.4·79-s − 0.550·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.30·7-s + 0.437·11-s − 1.63·13-s + 1.18·17-s + 0.917·19-s + 1.13·23-s − 0.800·25-s − 0.0187·29-s − 0.458·31-s − 0.583·35-s + 0.147·37-s − 1.85·41-s − 0.358·43-s + 0.926·47-s + 0.699·49-s − 1.22·53-s + 0.195·55-s + 1.86·59-s + 1.01·61-s − 0.731·65-s + 1.50·67-s − 0.925·71-s − 0.573·73-s − 0.569·77-s − 1.51·79-s − 0.0604·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 5 | \( 1 - T + 5T^{2} \) |
| 7 | \( 1 + 3.44T + 7T^{2} \) |
| 11 | \( 1 - 1.44T + 11T^{2} \) |
| 13 | \( 1 + 5.89T + 13T^{2} \) |
| 17 | \( 1 - 4.89T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 5.44T + 23T^{2} \) |
| 29 | \( 1 + 0.101T + 29T^{2} \) |
| 31 | \( 1 + 2.55T + 31T^{2} \) |
| 37 | \( 1 - 0.898T + 37T^{2} \) |
| 41 | \( 1 + 11.8T + 41T^{2} \) |
| 43 | \( 1 + 2.34T + 43T^{2} \) |
| 47 | \( 1 - 6.34T + 47T^{2} \) |
| 53 | \( 1 + 8.89T + 53T^{2} \) |
| 59 | \( 1 - 14.3T + 59T^{2} \) |
| 61 | \( 1 - 7.89T + 61T^{2} \) |
| 67 | \( 1 - 12.3T + 67T^{2} \) |
| 71 | \( 1 + 7.79T + 71T^{2} \) |
| 73 | \( 1 + 4.89T + 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 + 0.550T + 83T^{2} \) |
| 89 | \( 1 + 12.8T + 89T^{2} \) |
| 97 | \( 1 + 3.89T + 97T^{2} \) |
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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
−7.68453911928495929778195830349, −7.04582893240847866444465179953, −6.57700010281603526463336049804, −5.47490019602420060382957642251, −5.24651367239639846133098057128, −3.98189751968747500240525189500, −3.19762850076160979941288231547, −2.55475708146221426719369662537, −1.32450308330226505316492323893, 0,
1.32450308330226505316492323893, 2.55475708146221426719369662537, 3.19762850076160979941288231547, 3.98189751968747500240525189500, 5.24651367239639846133098057128, 5.47490019602420060382957642251, 6.57700010281603526463336049804, 7.04582893240847866444465179953, 7.68453911928495929778195830349
