L(s) = 1 | − 2-s − 2.96·3-s + 4-s + 2.96·6-s − 1.83·7-s − 8-s + 5.79·9-s − 1.83·11-s − 2.96·12-s + 2.41·13-s + 1.83·14-s + 16-s − 2.78·17-s − 5.79·18-s − 1.67·19-s + 5.43·21-s + 1.83·22-s + 7.76·23-s + 2.96·24-s − 2.41·26-s − 8.30·27-s − 1.83·28-s + 7.58·29-s − 5.29·31-s − 32-s + 5.43·33-s + 2.78·34-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.71·3-s + 0.5·4-s + 1.21·6-s − 0.692·7-s − 0.353·8-s + 1.93·9-s − 0.552·11-s − 0.856·12-s + 0.670·13-s + 0.489·14-s + 0.250·16-s − 0.675·17-s − 1.36·18-s − 0.383·19-s + 1.18·21-s + 0.390·22-s + 1.61·23-s + 0.605·24-s − 0.474·26-s − 1.59·27-s − 0.346·28-s + 1.40·29-s − 0.950·31-s − 0.176·32-s + 0.946·33-s + 0.477·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1250 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + 2.96T + 3T^{2} \) |
| 7 | \( 1 + 1.83T + 7T^{2} \) |
| 11 | \( 1 + 1.83T + 11T^{2} \) |
| 13 | \( 1 - 2.41T + 13T^{2} \) |
| 17 | \( 1 + 2.78T + 17T^{2} \) |
| 19 | \( 1 + 1.67T + 19T^{2} \) |
| 23 | \( 1 - 7.76T + 23T^{2} \) |
| 29 | \( 1 - 7.58T + 29T^{2} \) |
| 31 | \( 1 + 5.29T + 31T^{2} \) |
| 37 | \( 1 - 1.31T + 37T^{2} \) |
| 41 | \( 1 - 3.51T + 41T^{2} \) |
| 43 | \( 1 - 4.30T + 43T^{2} \) |
| 47 | \( 1 + 1.83T + 47T^{2} \) |
| 53 | \( 1 - 6.51T + 53T^{2} \) |
| 59 | \( 1 + 9.06T + 59T^{2} \) |
| 61 | \( 1 - 2.58T + 61T^{2} \) |
| 67 | \( 1 + 9.50T + 67T^{2} \) |
| 71 | \( 1 - 0.305T + 71T^{2} \) |
| 73 | \( 1 + 14.9T + 73T^{2} \) |
| 79 | \( 1 + 3.46T + 79T^{2} \) |
| 83 | \( 1 - 6.37T + 83T^{2} \) |
| 89 | \( 1 + 3.32T + 89T^{2} \) |
| 97 | \( 1 + 11.0T + 97T^{2} \) |
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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
−9.406748774186829580117080946510, −8.622665277242298101181247829012, −7.41609920287580038066767876219, −6.69010008057756855191289602995, −6.11848430634840757963624638758, −5.26545413621453236532688360002, −4.29670669709022366504859179522, −2.85399358228317768980031641938, −1.21064639138056017442617963411, 0,
1.21064639138056017442617963411, 2.85399358228317768980031641938, 4.29670669709022366504859179522, 5.26545413621453236532688360002, 6.11848430634840757963624638758, 6.69010008057756855191289602995, 7.41609920287580038066767876219, 8.622665277242298101181247829012, 9.406748774186829580117080946510