| L(s) = 1 | + 1.89·3-s − 2·4-s − 4.26·5-s + 0.586·9-s − 3.78·12-s − 8.07·15-s + 4·16-s + 8.52·20-s + 4.56·23-s + 13.1·25-s − 4.57·27-s + 10.5·31-s − 1.17·36-s + 11.2·37-s − 2.50·45-s − 13.6·47-s + 7.57·48-s − 7·49-s + 6.09·53-s + 12.3·59-s + 16.1·60-s − 8·64-s − 3.65·67-s + 8.64·69-s + 6.44·71-s + 24.9·75-s − 17.0·80-s + ⋯ |
| L(s) = 1 | + 1.09·3-s − 4-s − 1.90·5-s + 0.195·9-s − 1.09·12-s − 2.08·15-s + 16-s + 1.90·20-s + 0.951·23-s + 2.63·25-s − 0.879·27-s + 1.89·31-s − 0.195·36-s + 1.85·37-s − 0.372·45-s − 1.99·47-s + 1.09·48-s − 49-s + 0.837·53-s + 1.60·59-s + 2.08·60-s − 64-s − 0.445·67-s + 1.04·69-s + 0.764·71-s + 2.88·75-s − 1.90·80-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1331 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1331 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.161823730\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.161823730\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| good | 2 | \( 1 + 2T^{2} \) |
| 3 | \( 1 - 1.89T + 3T^{2} \) |
| 5 | \( 1 + 4.26T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 4.56T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 10.5T + 31T^{2} \) |
| 37 | \( 1 - 11.2T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 13.6T + 47T^{2} \) |
| 53 | \( 1 - 6.09T + 53T^{2} \) |
| 59 | \( 1 - 12.3T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 3.65T + 67T^{2} \) |
| 71 | \( 1 - 6.44T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 18.4T + 89T^{2} \) |
| 97 | \( 1 - 7.42T + 97T^{2} \) |
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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
−9.346893486413228164932501551273, −8.624548283347068413092090610583, −8.079508592499445541822681103371, −7.68159288850237310107866539048, −6.54033584772932814408017475777, −5.00565132828268278327436106876, −4.31835494713520340296366130861, −3.53118401985594889982810927561, −2.84866684099756572426694162065, −0.75023124058909643812044576344,
0.75023124058909643812044576344, 2.84866684099756572426694162065, 3.53118401985594889982810927561, 4.31835494713520340296366130861, 5.00565132828268278327436106876, 6.54033584772932814408017475777, 7.68159288850237310107866539048, 8.079508592499445541822681103371, 8.624548283347068413092090610583, 9.346893486413228164932501551273
