| L(s) = 1 | − 20.2·5-s + 24.7·7-s + 18.8·11-s − 48.4·13-s − 40.4·17-s − 7.82·19-s + 157.·23-s + 283.·25-s − 219.·29-s + 139.·31-s − 500.·35-s + 270.·37-s − 30.7·41-s − 57.2·43-s − 143.·47-s + 269.·49-s − 180.·53-s − 380.·55-s − 317.·59-s + 759.·61-s + 979.·65-s − 428.·67-s + 29.0·71-s − 327.·73-s + 465.·77-s + 1.02e3·79-s + 454.·83-s + ⋯ |
| L(s) = 1 | − 1.80·5-s + 1.33·7-s + 0.515·11-s − 1.03·13-s − 0.577·17-s − 0.0945·19-s + 1.43·23-s + 2.27·25-s − 1.40·29-s + 0.807·31-s − 2.41·35-s + 1.20·37-s − 0.117·41-s − 0.203·43-s − 0.444·47-s + 0.784·49-s − 0.466·53-s − 0.932·55-s − 0.699·59-s + 1.59·61-s + 1.86·65-s − 0.780·67-s + 0.0486·71-s − 0.525·73-s + 0.688·77-s + 1.45·79-s + 0.601·83-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
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 20.2T + 125T^{2} \) |
| 7 | \( 1 - 24.7T + 343T^{2} \) |
| 11 | \( 1 - 18.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 48.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 40.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 7.82T + 6.85e3T^{2} \) |
| 23 | \( 1 - 157.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 219.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 139.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 270.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 30.7T + 6.89e4T^{2} \) |
| 43 | \( 1 + 57.2T + 7.95e4T^{2} \) |
| 47 | \( 1 + 143.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 180.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 317.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 759.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 428.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 29.0T + 3.57e5T^{2} \) |
| 73 | \( 1 + 327.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.02e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 454.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 677.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 397.T + 9.12e5T^{2} \) |
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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
−8.681545883581886706735808475323, −7.967668594241124322424409005425, −7.44177976931052286402782445433, −6.68619775500511027354034591563, −5.13530945005946601749583766199, −4.56943020036617852735164622819, −3.82033759156381834900950035173, −2.63495305611533554371521289436, −1.21796822807513702656824736150, 0,
1.21796822807513702656824736150, 2.63495305611533554371521289436, 3.82033759156381834900950035173, 4.56943020036617852735164622819, 5.13530945005946601749583766199, 6.68619775500511027354034591563, 7.44177976931052286402782445433, 7.967668594241124322424409005425, 8.681545883581886706735808475323
