L(s) = 1 | − 2-s − 2.37·3-s + 4-s + 1.41·5-s + 2.37·6-s + 3.16·7-s − 8-s + 2.63·9-s − 1.41·10-s − 4.44·11-s − 2.37·12-s − 2.06·13-s − 3.16·14-s − 3.36·15-s + 16-s + 2.79·17-s − 2.63·18-s + 3.09·19-s + 1.41·20-s − 7.51·21-s + 4.44·22-s + 3.32·23-s + 2.37·24-s − 2.99·25-s + 2.06·26-s + 0.865·27-s + 3.16·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.37·3-s + 0.5·4-s + 0.633·5-s + 0.969·6-s + 1.19·7-s − 0.353·8-s + 0.878·9-s − 0.447·10-s − 1.33·11-s − 0.685·12-s − 0.571·13-s − 0.846·14-s − 0.868·15-s + 0.250·16-s + 0.678·17-s − 0.621·18-s + 0.710·19-s + 0.316·20-s − 1.64·21-s + 0.947·22-s + 0.692·23-s + 0.484·24-s − 0.598·25-s + 0.404·26-s + 0.166·27-s + 0.598·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 2011 | \( 1 + T \) |
good | 3 | \( 1 + 2.37T + 3T^{2} \) |
| 5 | \( 1 - 1.41T + 5T^{2} \) |
| 7 | \( 1 - 3.16T + 7T^{2} \) |
| 11 | \( 1 + 4.44T + 11T^{2} \) |
| 13 | \( 1 + 2.06T + 13T^{2} \) |
| 17 | \( 1 - 2.79T + 17T^{2} \) |
| 19 | \( 1 - 3.09T + 19T^{2} \) |
| 23 | \( 1 - 3.32T + 23T^{2} \) |
| 29 | \( 1 + 2.05T + 29T^{2} \) |
| 31 | \( 1 - 2.77T + 31T^{2} \) |
| 37 | \( 1 + 7.57T + 37T^{2} \) |
| 41 | \( 1 - 1.40T + 41T^{2} \) |
| 43 | \( 1 + 9.29T + 43T^{2} \) |
| 47 | \( 1 + 3.62T + 47T^{2} \) |
| 53 | \( 1 - 0.794T + 53T^{2} \) |
| 59 | \( 1 + 9.14T + 59T^{2} \) |
| 61 | \( 1 + 1.82T + 61T^{2} \) |
| 67 | \( 1 + 12.9T + 67T^{2} \) |
| 71 | \( 1 - 11.9T + 71T^{2} \) |
| 73 | \( 1 - 1.59T + 73T^{2} \) |
| 79 | \( 1 + 2.45T + 79T^{2} \) |
| 83 | \( 1 - 12.3T + 83T^{2} \) |
| 89 | \( 1 - 3.08T + 89T^{2} \) |
| 97 | \( 1 - 14.3T + 97T^{2} \) |
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
−7.87501092009975541074029141488, −7.52429351097212222238918709053, −6.57277880561600997145553244350, −5.76744458877357749339534779665, −5.09560414659095797667665363057, −4.88721071871327907670099159775, −3.21644122666525385387781363516, −2.10747152629033452399981909069, −1.23052197537468409620344145699, 0,
1.23052197537468409620344145699, 2.10747152629033452399981909069, 3.21644122666525385387781363516, 4.88721071871327907670099159775, 5.09560414659095797667665363057, 5.76744458877357749339534779665, 6.57277880561600997145553244350, 7.52429351097212222238918709053, 7.87501092009975541074029141488