L(s) = 1 | − 1.75·3-s + 5-s + 1.53·7-s + 0.0913·9-s − 5.86·11-s + 0.896·13-s − 1.75·15-s − 5.84·17-s + 3.50·19-s − 2.69·21-s + 2.74·23-s + 25-s + 5.11·27-s + 2.53·29-s + 5.38·31-s + 10.3·33-s + 1.53·35-s + 1.58·37-s − 1.57·39-s + 1.42·41-s + 8.60·43-s + 0.0913·45-s + 1.37·47-s − 4.65·49-s + 10.2·51-s − 10.7·53-s − 5.86·55-s + ⋯ |
L(s) = 1 | − 1.01·3-s + 0.447·5-s + 0.578·7-s + 0.0304·9-s − 1.76·11-s + 0.248·13-s − 0.453·15-s − 1.41·17-s + 0.804·19-s − 0.587·21-s + 0.571·23-s + 0.200·25-s + 0.984·27-s + 0.469·29-s + 0.966·31-s + 1.79·33-s + 0.258·35-s + 0.260·37-s − 0.252·39-s + 0.222·41-s + 1.31·43-s + 0.0136·45-s + 0.199·47-s − 0.664·49-s + 1.44·51-s − 1.47·53-s − 0.790·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 151 | \( 1 + T \) |
good | 3 | \( 1 + 1.75T + 3T^{2} \) |
| 7 | \( 1 - 1.53T + 7T^{2} \) |
| 11 | \( 1 + 5.86T + 11T^{2} \) |
| 13 | \( 1 - 0.896T + 13T^{2} \) |
| 17 | \( 1 + 5.84T + 17T^{2} \) |
| 19 | \( 1 - 3.50T + 19T^{2} \) |
| 23 | \( 1 - 2.74T + 23T^{2} \) |
| 29 | \( 1 - 2.53T + 29T^{2} \) |
| 31 | \( 1 - 5.38T + 31T^{2} \) |
| 37 | \( 1 - 1.58T + 37T^{2} \) |
| 41 | \( 1 - 1.42T + 41T^{2} \) |
| 43 | \( 1 - 8.60T + 43T^{2} \) |
| 47 | \( 1 - 1.37T + 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 59 | \( 1 + 0.677T + 59T^{2} \) |
| 61 | \( 1 - 8.86T + 61T^{2} \) |
| 67 | \( 1 + 9.39T + 67T^{2} \) |
| 71 | \( 1 + 4.08T + 71T^{2} \) |
| 73 | \( 1 + 4.48T + 73T^{2} \) |
| 79 | \( 1 - 6.93T + 79T^{2} \) |
| 83 | \( 1 + 5.69T + 83T^{2} \) |
| 89 | \( 1 + 3.48T + 89T^{2} \) |
| 97 | \( 1 + 10.1T + 97T^{2} \) |
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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
−7.72658892127455723797462730449, −6.90010849808164257648585863730, −6.16311759447646223802947221947, −5.54975049636331727317113496457, −4.92386575222181122895694284986, −4.46085672658379480264785109942, −2.97520130322760833077331801771, −2.40034872528875233770648228034, −1.14494982582733615719429592562, 0,
1.14494982582733615719429592562, 2.40034872528875233770648228034, 2.97520130322760833077331801771, 4.46085672658379480264785109942, 4.92386575222181122895694284986, 5.54975049636331727317113496457, 6.16311759447646223802947221947, 6.90010849808164257648585863730, 7.72658892127455723797462730449