| L(s) = 1 | − 1.37·3-s − 2.19·5-s + 1.41·7-s − 1.09·9-s + 2.07·11-s + 1.02·13-s + 3.02·15-s − 0.161·17-s − 19-s − 1.95·21-s − 5.93·23-s − 0.181·25-s + 5.65·27-s − 5.45·29-s + 1.39·31-s − 2.86·33-s − 3.11·35-s + 9.01·37-s − 1.41·39-s + 11.6·41-s − 7.88·43-s + 2.41·45-s + 0.333·47-s − 4.98·49-s + 0.222·51-s − 53-s − 4.56·55-s + ⋯ |
| L(s) = 1 | − 0.796·3-s − 0.981·5-s + 0.536·7-s − 0.366·9-s + 0.626·11-s + 0.284·13-s + 0.781·15-s − 0.0390·17-s − 0.229·19-s − 0.426·21-s − 1.23·23-s − 0.0362·25-s + 1.08·27-s − 1.01·29-s + 0.250·31-s − 0.499·33-s − 0.526·35-s + 1.48·37-s − 0.226·39-s + 1.82·41-s − 1.20·43-s + 0.359·45-s + 0.0487·47-s − 0.712·49-s + 0.0311·51-s − 0.137·53-s − 0.615·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8749578721\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8749578721\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + T \) |
| 53 | \( 1 + T \) |
| good | 3 | \( 1 + 1.37T + 3T^{2} \) |
| 5 | \( 1 + 2.19T + 5T^{2} \) |
| 7 | \( 1 - 1.41T + 7T^{2} \) |
| 11 | \( 1 - 2.07T + 11T^{2} \) |
| 13 | \( 1 - 1.02T + 13T^{2} \) |
| 17 | \( 1 + 0.161T + 17T^{2} \) |
| 23 | \( 1 + 5.93T + 23T^{2} \) |
| 29 | \( 1 + 5.45T + 29T^{2} \) |
| 31 | \( 1 - 1.39T + 31T^{2} \) |
| 37 | \( 1 - 9.01T + 37T^{2} \) |
| 41 | \( 1 - 11.6T + 41T^{2} \) |
| 43 | \( 1 + 7.88T + 43T^{2} \) |
| 47 | \( 1 - 0.333T + 47T^{2} \) |
| 59 | \( 1 + 3.93T + 59T^{2} \) |
| 61 | \( 1 - 3.52T + 61T^{2} \) |
| 67 | \( 1 + 2.78T + 67T^{2} \) |
| 71 | \( 1 + 6.57T + 71T^{2} \) |
| 73 | \( 1 - 0.117T + 73T^{2} \) |
| 79 | \( 1 - 1.80T + 79T^{2} \) |
| 83 | \( 1 + 5.99T + 83T^{2} \) |
| 89 | \( 1 - 0.233T + 89T^{2} \) |
| 97 | \( 1 - 6.34T + 97T^{2} \) |
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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
−8.189109758286595344077079814217, −7.88438590667598819508244027488, −6.94451460477743721945502961887, −6.10222141779770755140583161483, −5.62633079671501117332561775988, −4.50750786053294382768762159327, −4.09806243228200253182411101184, −3.07725341646394977655092974412, −1.81473241802779859560885343483, −0.55534624265046661360058914905,
0.55534624265046661360058914905, 1.81473241802779859560885343483, 3.07725341646394977655092974412, 4.09806243228200253182411101184, 4.50750786053294382768762159327, 5.62633079671501117332561775988, 6.10222141779770755140583161483, 6.94451460477743721945502961887, 7.88438590667598819508244027488, 8.189109758286595344077079814217
