| L(s) = 1 | − 4.07·2-s + 8.61·4-s − 13.3·7-s − 2.51·8-s − 11.5·11-s + 40.0·13-s + 54.3·14-s − 58.6·16-s + 93.3·17-s + 75.1·19-s + 47.1·22-s − 142.·23-s − 163.·26-s − 114.·28-s + 174.·29-s + 248.·31-s + 259.·32-s − 380.·34-s − 82.9·37-s − 306.·38-s + 449.·41-s − 279.·43-s − 99.6·44-s + 578.·46-s + 33.8·47-s − 164.·49-s + 345.·52-s + ⋯ |
| L(s) = 1 | − 1.44·2-s + 1.07·4-s − 0.720·7-s − 0.110·8-s − 0.316·11-s + 0.855·13-s + 1.03·14-s − 0.917·16-s + 1.33·17-s + 0.906·19-s + 0.456·22-s − 1.28·23-s − 1.23·26-s − 0.775·28-s + 1.11·29-s + 1.44·31-s + 1.43·32-s − 1.91·34-s − 0.368·37-s − 1.30·38-s + 1.71·41-s − 0.992·43-s − 0.341·44-s + 1.85·46-s + 0.105·47-s − 0.480·49-s + 0.920·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9279648203\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9279648203\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 4.07T + 8T^{2} \) |
| 7 | \( 1 + 13.3T + 343T^{2} \) |
| 11 | \( 1 + 11.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 40.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 93.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 75.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 142.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 174.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 248.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 82.9T + 5.06e4T^{2} \) |
| 41 | \( 1 - 449.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 279.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 33.8T + 1.03e5T^{2} \) |
| 53 | \( 1 + 423.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 615.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 502.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 57.7T + 3.00e5T^{2} \) |
| 71 | \( 1 - 252.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 823.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 205.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 646.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.32e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 563.T + 9.12e5T^{2} \) |
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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
−8.792568781995729420183802678567, −8.035559314162226479399864735724, −7.62026208253820051851088961053, −6.55028493394276490940763508554, −5.95674387080457207428185524510, −4.78418620823425874043642028363, −3.59667594526993929640380390601, −2.69037219139650084357239777171, −1.40082793166661906214909051039, −0.60610105440904540375054898335,
0.60610105440904540375054898335, 1.40082793166661906214909051039, 2.69037219139650084357239777171, 3.59667594526993929640380390601, 4.78418620823425874043642028363, 5.95674387080457207428185524510, 6.55028493394276490940763508554, 7.62026208253820051851088961053, 8.035559314162226479399864735724, 8.792568781995729420183802678567
