| L(s) = 1 | − 2·2-s + 4·4-s − 23·7-s − 8·8-s − 30·11-s + 34·13-s + 46·14-s + 16·16-s − 42·17-s − 139·19-s + 60·22-s + 192·23-s − 68·26-s − 92·28-s − 234·29-s − 55·31-s − 32·32-s + 84·34-s − 191·37-s + 278·38-s − 138·41-s − 53·43-s − 120·44-s − 384·46-s + 366·47-s + 186·49-s + 136·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.24·7-s − 0.353·8-s − 0.822·11-s + 0.725·13-s + 0.878·14-s + 1/4·16-s − 0.599·17-s − 1.67·19-s + 0.581·22-s + 1.74·23-s − 0.512·26-s − 0.620·28-s − 1.49·29-s − 0.318·31-s − 0.176·32-s + 0.423·34-s − 0.848·37-s + 1.18·38-s − 0.525·41-s − 0.187·43-s − 0.411·44-s − 1.23·46-s + 1.13·47-s + 0.542·49-s + 0.362·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6276504436\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6276504436\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 23 T + p^{3} T^{2} \) |
| 11 | \( 1 + 30 T + p^{3} T^{2} \) |
| 13 | \( 1 - 34 T + p^{3} T^{2} \) |
| 17 | \( 1 + 42 T + p^{3} T^{2} \) |
| 19 | \( 1 + 139 T + p^{3} T^{2} \) |
| 23 | \( 1 - 192 T + p^{3} T^{2} \) |
| 29 | \( 1 + 234 T + p^{3} T^{2} \) |
| 31 | \( 1 + 55 T + p^{3} T^{2} \) |
| 37 | \( 1 + 191 T + p^{3} T^{2} \) |
| 41 | \( 1 + 138 T + p^{3} T^{2} \) |
| 43 | \( 1 + 53 T + p^{3} T^{2} \) |
| 47 | \( 1 - 366 T + p^{3} T^{2} \) |
| 53 | \( 1 + 330 T + p^{3} T^{2} \) |
| 59 | \( 1 - 396 T + p^{3} T^{2} \) |
| 61 | \( 1 - 23 T + p^{3} T^{2} \) |
| 67 | \( 1 + 452 T + p^{3} T^{2} \) |
| 71 | \( 1 + 204 T + p^{3} T^{2} \) |
| 73 | \( 1 - 691 T + p^{3} T^{2} \) |
| 79 | \( 1 + 709 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1098 T + p^{3} T^{2} \) |
| 89 | \( 1 - 816 T + p^{3} T^{2} \) |
| 97 | \( 1 + 905 T + p^{3} T^{2} \) |
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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
−9.020414719493147599201199996626, −8.752796885187632354889839943056, −7.61290891017786948382709516660, −6.79542119730897351272597990942, −6.19935411646581590145207241678, −5.18266651038922849968863820836, −3.86410324693275203272135440802, −2.96282001380703127346449455736, −1.92910844111098546144712271567, −0.41922779125708855561555813045,
0.41922779125708855561555813045, 1.92910844111098546144712271567, 2.96282001380703127346449455736, 3.86410324693275203272135440802, 5.18266651038922849968863820836, 6.19935411646581590145207241678, 6.79542119730897351272597990942, 7.61290891017786948382709516660, 8.752796885187632354889839943056, 9.020414719493147599201199996626
