| L(s) = 1 | − 0.302·3-s − 3·5-s − 7-s − 2.90·9-s + 0.697·11-s − 5.60·13-s + 0.908·15-s − 5.30·17-s − 19-s + 0.302·21-s + 3·23-s + 4·25-s + 1.78·27-s + 9.90·29-s − 1.30·31-s − 0.211·33-s + 3·35-s + 3.60·37-s + 1.69·39-s + 0.697·41-s + 10·43-s + 8.72·45-s − 6.21·47-s + 49-s + 1.60·51-s − 6.90·53-s − 2.09·55-s + ⋯ |
| L(s) = 1 | − 0.174·3-s − 1.34·5-s − 0.377·7-s − 0.969·9-s + 0.210·11-s − 1.55·13-s + 0.234·15-s − 1.28·17-s − 0.229·19-s + 0.0660·21-s + 0.625·23-s + 0.800·25-s + 0.344·27-s + 1.83·29-s − 0.233·31-s − 0.0367·33-s + 0.507·35-s + 0.592·37-s + 0.271·39-s + 0.108·41-s + 1.52·43-s + 1.30·45-s − 0.905·47-s + 0.142·49-s + 0.224·51-s − 0.948·53-s − 0.282·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2128 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5755862034\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5755862034\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 0.302T + 3T^{2} \) |
| 5 | \( 1 + 3T + 5T^{2} \) |
| 11 | \( 1 - 0.697T + 11T^{2} \) |
| 13 | \( 1 + 5.60T + 13T^{2} \) |
| 17 | \( 1 + 5.30T + 17T^{2} \) |
| 23 | \( 1 - 3T + 23T^{2} \) |
| 29 | \( 1 - 9.90T + 29T^{2} \) |
| 31 | \( 1 + 1.30T + 31T^{2} \) |
| 37 | \( 1 - 3.60T + 37T^{2} \) |
| 41 | \( 1 - 0.697T + 41T^{2} \) |
| 43 | \( 1 - 10T + 43T^{2} \) |
| 47 | \( 1 + 6.21T + 47T^{2} \) |
| 53 | \( 1 + 6.90T + 53T^{2} \) |
| 59 | \( 1 - 6.21T + 59T^{2} \) |
| 61 | \( 1 + 4.21T + 61T^{2} \) |
| 67 | \( 1 - 1.90T + 67T^{2} \) |
| 71 | \( 1 + 12.2T + 71T^{2} \) |
| 73 | \( 1 - 1.51T + 73T^{2} \) |
| 79 | \( 1 + 11.2T + 79T^{2} \) |
| 83 | \( 1 - 12.9T + 83T^{2} \) |
| 89 | \( 1 + 10.6T + 89T^{2} \) |
| 97 | \( 1 - 9.60T + 97T^{2} \) |
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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
−8.948756209383605109634339802010, −8.340133643344082538301705693562, −7.51225691220439603161741445514, −6.85339261656809563553101462154, −6.03830756518747768235414161385, −4.82210746950053998768951198540, −4.37733556261980518585125553592, −3.18334415725561606049348262470, −2.46347910014738945302711450530, −0.47210294374268222490726203201,
0.47210294374268222490726203201, 2.46347910014738945302711450530, 3.18334415725561606049348262470, 4.37733556261980518585125553592, 4.82210746950053998768951198540, 6.03830756518747768235414161385, 6.85339261656809563553101462154, 7.51225691220439603161741445514, 8.340133643344082538301705693562, 8.948756209383605109634339802010
