L(s) = 1 | − 1.45·2-s − 3-s + 0.119·4-s + 2.38·5-s + 1.45·6-s + 7-s + 2.73·8-s + 9-s − 3.47·10-s − 0.119·12-s + 0.911·13-s − 1.45·14-s − 2.38·15-s − 4.22·16-s + 2.18·17-s − 1.45·18-s − 0.711·19-s + 0.285·20-s − 21-s − 7.80·23-s − 2.73·24-s + 0.698·25-s − 1.32·26-s − 27-s + 0.119·28-s − 8.86·29-s + 3.47·30-s + ⋯ |
L(s) = 1 | − 1.02·2-s − 0.577·3-s + 0.0598·4-s + 1.06·5-s + 0.594·6-s + 0.377·7-s + 0.967·8-s + 0.333·9-s − 1.09·10-s − 0.0345·12-s + 0.252·13-s − 0.389·14-s − 0.616·15-s − 1.05·16-s + 0.530·17-s − 0.343·18-s − 0.163·19-s + 0.0638·20-s − 0.218·21-s − 1.62·23-s − 0.558·24-s + 0.139·25-s − 0.260·26-s − 0.192·27-s + 0.0226·28-s − 1.64·29-s + 0.634·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 1.45T + 2T^{2} \) |
| 5 | \( 1 - 2.38T + 5T^{2} \) |
| 13 | \( 1 - 0.911T + 13T^{2} \) |
| 17 | \( 1 - 2.18T + 17T^{2} \) |
| 19 | \( 1 + 0.711T + 19T^{2} \) |
| 23 | \( 1 + 7.80T + 23T^{2} \) |
| 29 | \( 1 + 8.86T + 29T^{2} \) |
| 31 | \( 1 + 6.62T + 31T^{2} \) |
| 37 | \( 1 + 4.20T + 37T^{2} \) |
| 41 | \( 1 - 11.9T + 41T^{2} \) |
| 43 | \( 1 + 7.51T + 43T^{2} \) |
| 47 | \( 1 - 6.27T + 47T^{2} \) |
| 53 | \( 1 + 6.56T + 53T^{2} \) |
| 59 | \( 1 + 9.62T + 59T^{2} \) |
| 61 | \( 1 - 0.0627T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 + 5.96T + 71T^{2} \) |
| 73 | \( 1 - 4.38T + 73T^{2} \) |
| 79 | \( 1 - 4.85T + 79T^{2} \) |
| 83 | \( 1 + 10.0T + 83T^{2} \) |
| 89 | \( 1 + 0.261T + 89T^{2} \) |
| 97 | \( 1 - 9.88T + 97T^{2} \) |
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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
−8.668408313641529526464205625994, −7.76458940364379208751688762937, −7.26048853503182527950137730048, −6.02000741591985448199994507222, −5.67412788738642805619783423840, −4.65865422873028976873522234288, −3.73311222994139447208281275080, −2.05800354000012533981758861436, −1.47411492963424938030330719875, 0,
1.47411492963424938030330719875, 2.05800354000012533981758861436, 3.73311222994139447208281275080, 4.65865422873028976873522234288, 5.67412788738642805619783423840, 6.02000741591985448199994507222, 7.26048853503182527950137730048, 7.76458940364379208751688762937, 8.668408313641529526464205625994