| L(s) = 1 | − 5.99·5-s − 15.5·7-s − 34.8·11-s − 80.6·13-s − 70.1·17-s + 4.25·19-s − 118.·23-s − 89.0·25-s + 123.·29-s + 185.·31-s + 93.3·35-s + 151.·37-s + 212.·41-s − 290.·43-s + 212.·47-s − 100.·49-s − 556.·53-s + 209.·55-s + 853.·59-s − 688.·61-s + 483.·65-s + 915.·67-s − 786.·71-s − 993.·73-s + 543.·77-s + 568.·79-s + 747.·83-s + ⋯ |
| L(s) = 1 | − 0.536·5-s − 0.841·7-s − 0.956·11-s − 1.72·13-s − 1.00·17-s + 0.0513·19-s − 1.07·23-s − 0.712·25-s + 0.790·29-s + 1.07·31-s + 0.450·35-s + 0.673·37-s + 0.809·41-s − 1.02·43-s + 0.659·47-s − 0.292·49-s − 1.44·53-s + 0.512·55-s + 1.88·59-s − 1.44·61-s + 0.922·65-s + 1.66·67-s − 1.31·71-s − 1.59·73-s + 0.804·77-s + 0.809·79-s + 0.987·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4840550977\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4840550977\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 5.99T + 125T^{2} \) |
| 7 | \( 1 + 15.5T + 343T^{2} \) |
| 11 | \( 1 + 34.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 80.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 70.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 4.25T + 6.85e3T^{2} \) |
| 23 | \( 1 + 118.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 123.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 185.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 151.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 212.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 290.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 212.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 556.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 853.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 688.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 915.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 786.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 993.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 568.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 747.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.01e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.21e3T + 9.12e5T^{2} \) |
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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
−9.516171358796272324197126054261, −8.294555175383088795614254959853, −7.71547299028531919141740144728, −6.86725384568123500840375783809, −6.03825119578950386742472813584, −4.92388872884699414186969772677, −4.21010312836848021237484981351, −2.96739829218898489470898038379, −2.24286290815623580933120733424, −0.32454447779393009031655204773,
0.32454447779393009031655204773, 2.24286290815623580933120733424, 2.96739829218898489470898038379, 4.21010312836848021237484981351, 4.92388872884699414186969772677, 6.03825119578950386742472813584, 6.86725384568123500840375783809, 7.71547299028531919141740144728, 8.294555175383088795614254959853, 9.516171358796272324197126054261
