| L(s) = 1 | − 5·2-s + 3·3-s + 17·4-s − 15·6-s − 16·7-s − 45·8-s + 9·9-s − 44·11-s + 51·12-s − 78·13-s + 80·14-s + 89·16-s − 18·17-s − 45·18-s − 28·19-s − 48·21-s + 220·22-s − 184·23-s − 135·24-s + 390·26-s + 27·27-s − 272·28-s + 29·29-s − 224·31-s − 85·32-s − 132·33-s + 90·34-s + ⋯ |
| L(s) = 1 | − 1.76·2-s + 0.577·3-s + 17/8·4-s − 1.02·6-s − 0.863·7-s − 1.98·8-s + 1/3·9-s − 1.20·11-s + 1.22·12-s − 1.66·13-s + 1.52·14-s + 1.39·16-s − 0.256·17-s − 0.589·18-s − 0.338·19-s − 0.498·21-s + 2.13·22-s − 1.66·23-s − 1.14·24-s + 2.94·26-s + 0.192·27-s − 1.83·28-s + 0.185·29-s − 1.29·31-s − 0.469·32-s − 0.696·33-s + 0.453·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - p T \) |
| 5 | \( 1 \) |
| 29 | \( 1 - p T \) |
| good | 2 | \( 1 + 5 T + p^{3} T^{2} \) |
| 7 | \( 1 + 16 T + p^{3} T^{2} \) |
| 11 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 13 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 17 | \( 1 + 18 T + p^{3} T^{2} \) |
| 19 | \( 1 + 28 T + p^{3} T^{2} \) |
| 23 | \( 1 + 8 p T + p^{3} T^{2} \) |
| 31 | \( 1 + 224 T + p^{3} T^{2} \) |
| 37 | \( 1 + 254 T + p^{3} T^{2} \) |
| 41 | \( 1 + 78 T + p^{3} T^{2} \) |
| 43 | \( 1 - 260 T + p^{3} T^{2} \) |
| 47 | \( 1 + 312 T + p^{3} T^{2} \) |
| 53 | \( 1 + 574 T + p^{3} T^{2} \) |
| 59 | \( 1 - 180 T + p^{3} T^{2} \) |
| 61 | \( 1 + 10 p T + p^{3} T^{2} \) |
| 67 | \( 1 - 340 T + p^{3} T^{2} \) |
| 71 | \( 1 - 296 T + p^{3} T^{2} \) |
| 73 | \( 1 + 394 T + p^{3} T^{2} \) |
| 79 | \( 1 + 960 T + p^{3} T^{2} \) |
| 83 | \( 1 - 908 T + p^{3} T^{2} \) |
| 89 | \( 1 + 990 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1234 T + p^{3} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.994998785593754847490276583999, −7.47924803281431699050672798748, −6.83391067655151459153220320592, −5.88951752091034193698067128248, −4.71678990498982780687333634282, −3.31739973970105445144356077472, −2.44968371113395567636761170728, −1.83528885658038055157024328758, 0, 0,
1.83528885658038055157024328758, 2.44968371113395567636761170728, 3.31739973970105445144356077472, 4.71678990498982780687333634282, 5.88951752091034193698067128248, 6.83391067655151459153220320592, 7.47924803281431699050672798748, 7.994998785593754847490276583999
