| L(s) = 1 | − 3-s + 5-s + 3.78·7-s + 9-s + 3.67·11-s − 6.64·13-s − 15-s − 1.58·17-s + 6.04·19-s − 3.78·21-s + 0.537·23-s + 25-s − 27-s + 4.44·29-s + 3.41·31-s − 3.67·33-s + 3.78·35-s − 9.04·37-s + 6.64·39-s − 10.9·41-s + 2.22·43-s + 45-s − 0.535·47-s + 7.30·49-s + 1.58·51-s + 3.27·53-s + 3.67·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.42·7-s + 0.333·9-s + 1.10·11-s − 1.84·13-s − 0.258·15-s − 0.384·17-s + 1.38·19-s − 0.825·21-s + 0.112·23-s + 0.200·25-s − 0.192·27-s + 0.825·29-s + 0.612·31-s − 0.640·33-s + 0.639·35-s − 1.48·37-s + 1.06·39-s − 1.70·41-s + 0.339·43-s + 0.149·45-s − 0.0781·47-s + 1.04·49-s + 0.222·51-s + 0.449·53-s + 0.496·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.329908012\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.329908012\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 67 | \( 1 + T \) |
| good | 7 | \( 1 - 3.78T + 7T^{2} \) |
| 11 | \( 1 - 3.67T + 11T^{2} \) |
| 13 | \( 1 + 6.64T + 13T^{2} \) |
| 17 | \( 1 + 1.58T + 17T^{2} \) |
| 19 | \( 1 - 6.04T + 19T^{2} \) |
| 23 | \( 1 - 0.537T + 23T^{2} \) |
| 29 | \( 1 - 4.44T + 29T^{2} \) |
| 31 | \( 1 - 3.41T + 31T^{2} \) |
| 37 | \( 1 + 9.04T + 37T^{2} \) |
| 41 | \( 1 + 10.9T + 41T^{2} \) |
| 43 | \( 1 - 2.22T + 43T^{2} \) |
| 47 | \( 1 + 0.535T + 47T^{2} \) |
| 53 | \( 1 - 3.27T + 53T^{2} \) |
| 59 | \( 1 - 9.88T + 59T^{2} \) |
| 61 | \( 1 - 15.4T + 61T^{2} \) |
| 71 | \( 1 - 9.02T + 71T^{2} \) |
| 73 | \( 1 - 7.14T + 73T^{2} \) |
| 79 | \( 1 + 4.77T + 79T^{2} \) |
| 83 | \( 1 + 3.54T + 83T^{2} \) |
| 89 | \( 1 - 13.4T + 89T^{2} \) |
| 97 | \( 1 - 10.9T + 97T^{2} \) |
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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
−7.76259507377682299008623130070, −6.94939984348782343724726187091, −6.69803890286122165673826715461, −5.43183342091205453287125107340, −5.12442271981707728144371298690, −4.57438139316038227038327766485, −3.61408352835896533423185709472, −2.44802889631829826118380434329, −1.70723279140214842566028830479, −0.818402482245154783180317386507,
0.818402482245154783180317386507, 1.70723279140214842566028830479, 2.44802889631829826118380434329, 3.61408352835896533423185709472, 4.57438139316038227038327766485, 5.12442271981707728144371298690, 5.43183342091205453287125107340, 6.69803890286122165673826715461, 6.94939984348782343724726187091, 7.76259507377682299008623130070
