| L(s) = 1 | − 5·5-s − 19.1·7-s − 18.3·11-s + 14.0·13-s + 63.8·17-s + 101.·19-s + 155.·23-s + 25·25-s − 58.2·29-s + 136.·31-s + 95.9·35-s − 117.·37-s − 276.·41-s − 451.·43-s − 15.2·47-s + 25.0·49-s − 266.·53-s + 91.9·55-s + 585.·59-s − 392.·61-s − 70.4·65-s − 1.03e3·67-s − 657.·71-s + 125.·73-s + 352.·77-s − 962.·79-s + 721.·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.03·7-s − 0.503·11-s + 0.300·13-s + 0.911·17-s + 1.22·19-s + 1.40·23-s + 0.200·25-s − 0.372·29-s + 0.790·31-s + 0.463·35-s − 0.523·37-s − 1.05·41-s − 1.60·43-s − 0.0474·47-s + 0.0729·49-s − 0.691·53-s + 0.225·55-s + 1.29·59-s − 0.823·61-s − 0.134·65-s − 1.88·67-s − 1.09·71-s + 0.201·73-s + 0.522·77-s − 1.37·79-s + 0.953·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 5T \) |
| good | 7 | \( 1 + 19.1T + 343T^{2} \) |
| 11 | \( 1 + 18.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 14.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 63.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 101.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 155.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 58.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 136.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 117.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 276.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 451.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 15.2T + 1.03e5T^{2} \) |
| 53 | \( 1 + 266.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 585.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 392.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 1.03e3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 657.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 125.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 962.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 721.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 122.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 153.T + 9.12e5T^{2} \) |
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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
−9.146765332095930263846111194468, −8.240000244353871589437763552647, −7.37045127062988763165137841770, −6.66661620627109873543208028839, −5.62289566389347645539522540398, −4.80343809108110059144737602975, −3.40606131883157535961881430756, −3.02591750431436536117035434708, −1.26785452361844842505032605540, 0,
1.26785452361844842505032605540, 3.02591750431436536117035434708, 3.40606131883157535961881430756, 4.80343809108110059144737602975, 5.62289566389347645539522540398, 6.66661620627109873543208028839, 7.37045127062988763165137841770, 8.240000244353871589437763552647, 9.146765332095930263846111194468
