L(s) = 1 | + 3-s + 1.32·5-s − 7-s + 9-s − 3.74·11-s − 5.07·13-s + 1.32·15-s + 3.38·17-s + 3.74·19-s − 21-s + 23-s − 3.23·25-s + 27-s − 8.01·29-s + 10.2·31-s − 3.74·33-s − 1.32·35-s − 0.911·37-s − 5.07·39-s − 7.38·41-s + 4.08·43-s + 1.32·45-s − 0.831·47-s + 49-s + 3.38·51-s + 10.0·53-s − 4.96·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.593·5-s − 0.377·7-s + 0.333·9-s − 1.12·11-s − 1.40·13-s + 0.342·15-s + 0.820·17-s + 0.858·19-s − 0.218·21-s + 0.208·23-s − 0.647·25-s + 0.192·27-s − 1.48·29-s + 1.83·31-s − 0.651·33-s − 0.224·35-s − 0.149·37-s − 0.811·39-s − 1.15·41-s + 0.622·43-s + 0.197·45-s − 0.121·47-s + 0.142·49-s + 0.473·51-s + 1.38·53-s − 0.669·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 - 1.32T + 5T^{2} \) |
| 11 | \( 1 + 3.74T + 11T^{2} \) |
| 13 | \( 1 + 5.07T + 13T^{2} \) |
| 17 | \( 1 - 3.38T + 17T^{2} \) |
| 19 | \( 1 - 3.74T + 19T^{2} \) |
| 29 | \( 1 + 8.01T + 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 + 0.911T + 37T^{2} \) |
| 41 | \( 1 + 7.38T + 41T^{2} \) |
| 43 | \( 1 - 4.08T + 43T^{2} \) |
| 47 | \( 1 + 0.831T + 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 + 6.42T + 59T^{2} \) |
| 61 | \( 1 - 2.86T + 61T^{2} \) |
| 67 | \( 1 + 2.70T + 67T^{2} \) |
| 71 | \( 1 + 0.210T + 71T^{2} \) |
| 73 | \( 1 + 12.8T + 73T^{2} \) |
| 79 | \( 1 + 10.2T + 79T^{2} \) |
| 83 | \( 1 - 1.20T + 83T^{2} \) |
| 89 | \( 1 + 14.8T + 89T^{2} \) |
| 97 | \( 1 + 12.8T + 97T^{2} \) |
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
−7.48458646483744408839370175005, −7.09528812815843418153476615235, −6.00035724978743530936790746270, −5.41175101172213511998256277433, −4.81214804430731404100551257584, −3.80700374528165022178462090810, −2.86020545742159243800374438650, −2.49217517576195735640266702170, −1.41482446602480954712250837880, 0,
1.41482446602480954712250837880, 2.49217517576195735640266702170, 2.86020545742159243800374438650, 3.80700374528165022178462090810, 4.81214804430731404100551257584, 5.41175101172213511998256277433, 6.00035724978743530936790746270, 7.09528812815843418153476615235, 7.48458646483744408839370175005