| L(s) = 1 | − 5·5-s + 11.2·7-s − 61.3·11-s − 75.7·13-s − 11.1·17-s + 71.5·19-s − 126.·23-s + 25·25-s + 235.·29-s + 110.·31-s − 56.4·35-s + 434.·37-s + 1.15·41-s − 77.6·43-s + 231.·47-s − 215.·49-s − 500.·53-s + 306.·55-s + 334.·59-s + 147.·61-s + 378.·65-s + 84.9·67-s + 101.·71-s − 50.4·73-s − 693.·77-s − 818.·79-s − 206.·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.610·7-s − 1.68·11-s − 1.61·13-s − 0.159·17-s + 0.863·19-s − 1.14·23-s + 0.200·25-s + 1.50·29-s + 0.642·31-s − 0.272·35-s + 1.92·37-s + 0.00441·41-s − 0.275·43-s + 0.718·47-s − 0.627·49-s − 1.29·53-s + 0.752·55-s + 0.738·59-s + 0.308·61-s + 0.722·65-s + 0.154·67-s + 0.169·71-s − 0.0809·73-s − 1.02·77-s − 1.16·79-s − 0.272·83-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
| \(L(2)\) |
\(\approx\) |
\(1.335229840\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.335229840\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 11.2T + 343T^{2} \) |
| 11 | \( 1 + 61.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 75.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 11.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 71.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 126.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 235.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 110.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 434.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 1.15T + 6.89e4T^{2} \) |
| 43 | \( 1 + 77.6T + 7.95e4T^{2} \) |
| 47 | \( 1 - 231.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 500.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 334.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 147.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 84.9T + 3.00e5T^{2} \) |
| 71 | \( 1 - 101.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 50.4T + 3.89e5T^{2} \) |
| 79 | \( 1 + 818.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 206.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 648.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.88e3T + 9.12e5T^{2} \) |
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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
−9.773729882504266493661166332454, −8.400209607619797325737578265073, −7.82941294829324340917537225658, −7.29697958181257840627618900075, −6.02535721891704731312272546076, −4.96251707276494288359305420012, −4.54336358652557692552680739202, −2.99769819163523705684466175790, −2.24531282170143901190608403049, −0.58136826099589294356913303943,
0.58136826099589294356913303943, 2.24531282170143901190608403049, 2.99769819163523705684466175790, 4.54336358652557692552680739202, 4.96251707276494288359305420012, 6.02535721891704731312272546076, 7.29697958181257840627618900075, 7.82941294829324340917537225658, 8.400209607619797325737578265073, 9.773729882504266493661166332454
