L(s) = 1 | − 0.183·3-s + 1.35·7-s − 2.96·9-s − 2.79·11-s − 2.49·13-s + 4.95·17-s + 5.28·19-s − 0.248·21-s − 7.44·23-s + 1.09·27-s + 0.314·29-s + 5.26·31-s + 0.512·33-s + 0.757·37-s + 0.456·39-s − 1.20·41-s + 10.3·43-s − 4.27·47-s − 5.15·49-s − 0.908·51-s − 13.0·53-s − 0.968·57-s + 5.02·59-s − 4.73·61-s − 4.03·63-s + 13.5·67-s + 1.36·69-s + ⋯ |
L(s) = 1 | − 0.105·3-s + 0.513·7-s − 0.988·9-s − 0.844·11-s − 0.691·13-s + 1.20·17-s + 1.21·19-s − 0.0543·21-s − 1.55·23-s + 0.210·27-s + 0.0584·29-s + 0.944·31-s + 0.0892·33-s + 0.124·37-s + 0.0731·39-s − 0.187·41-s + 1.57·43-s − 0.623·47-s − 0.736·49-s − 0.127·51-s − 1.79·53-s − 0.128·57-s + 0.653·59-s − 0.605·61-s − 0.507·63-s + 1.65·67-s + 0.164·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
good | 3 | \( 1 + 0.183T + 3T^{2} \) |
| 7 | \( 1 - 1.35T + 7T^{2} \) |
| 11 | \( 1 + 2.79T + 11T^{2} \) |
| 13 | \( 1 + 2.49T + 13T^{2} \) |
| 17 | \( 1 - 4.95T + 17T^{2} \) |
| 19 | \( 1 - 5.28T + 19T^{2} \) |
| 23 | \( 1 + 7.44T + 23T^{2} \) |
| 29 | \( 1 - 0.314T + 29T^{2} \) |
| 31 | \( 1 - 5.26T + 31T^{2} \) |
| 37 | \( 1 - 0.757T + 37T^{2} \) |
| 41 | \( 1 + 1.20T + 41T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 + 4.27T + 47T^{2} \) |
| 53 | \( 1 + 13.0T + 53T^{2} \) |
| 59 | \( 1 - 5.02T + 59T^{2} \) |
| 61 | \( 1 + 4.73T + 61T^{2} \) |
| 67 | \( 1 - 13.5T + 67T^{2} \) |
| 71 | \( 1 + 0.185T + 71T^{2} \) |
| 73 | \( 1 - 9.89T + 73T^{2} \) |
| 79 | \( 1 - 16.3T + 79T^{2} \) |
| 83 | \( 1 + 12.1T + 83T^{2} \) |
| 89 | \( 1 + 12.6T + 89T^{2} \) |
| 97 | \( 1 + 10.2T + 97T^{2} \) |
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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
−7.57780516242416217218183162679, −6.56993457965940524048502279932, −5.80408790658347787900557528417, −5.29499229188461869291403546097, −4.75906343724850125787179569524, −3.71513194746636352285778659819, −2.93124544345099681874774497643, −2.29900573311945723566364674924, −1.15468722416862480228470721792, 0,
1.15468722416862480228470721792, 2.29900573311945723566364674924, 2.93124544345099681874774497643, 3.71513194746636352285778659819, 4.75906343724850125787179569524, 5.29499229188461869291403546097, 5.80408790658347787900557528417, 6.56993457965940524048502279932, 7.57780516242416217218183162679