| L(s) = 1 | − 2.63·2-s + 2.69·3-s + 4.93·4-s − 7.10·6-s + 2.74·7-s − 7.71·8-s + 4.28·9-s + 1.84·11-s + 13.3·12-s − 7.21·14-s + 10.4·16-s + 3.81·17-s − 11.2·18-s − 5.67·19-s + 7.39·21-s − 4.84·22-s + 0.139·23-s − 20.8·24-s + 3.46·27-s + 13.5·28-s + 1.16·29-s − 5.69·31-s − 12.0·32-s + 4.96·33-s − 10.0·34-s + 21.1·36-s + 2.46·37-s + ⋯ |
| L(s) = 1 | − 1.86·2-s + 1.55·3-s + 2.46·4-s − 2.90·6-s + 1.03·7-s − 2.72·8-s + 1.42·9-s + 0.554·11-s + 3.84·12-s − 1.92·14-s + 2.61·16-s + 0.925·17-s − 2.65·18-s − 1.30·19-s + 1.61·21-s − 1.03·22-s + 0.0290·23-s − 4.24·24-s + 0.667·27-s + 2.55·28-s + 0.216·29-s − 1.02·31-s − 2.13·32-s + 0.864·33-s − 1.72·34-s + 3.52·36-s + 0.404·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.810206654\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.810206654\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.63T + 2T^{2} \) |
| 3 | \( 1 - 2.69T + 3T^{2} \) |
| 7 | \( 1 - 2.74T + 7T^{2} \) |
| 11 | \( 1 - 1.84T + 11T^{2} \) |
| 17 | \( 1 - 3.81T + 17T^{2} \) |
| 19 | \( 1 + 5.67T + 19T^{2} \) |
| 23 | \( 1 - 0.139T + 23T^{2} \) |
| 29 | \( 1 - 1.16T + 29T^{2} \) |
| 31 | \( 1 + 5.69T + 31T^{2} \) |
| 37 | \( 1 - 2.46T + 37T^{2} \) |
| 41 | \( 1 - 9.25T + 41T^{2} \) |
| 43 | \( 1 - 2.09T + 43T^{2} \) |
| 47 | \( 1 - 3.66T + 47T^{2} \) |
| 53 | \( 1 - 9.97T + 53T^{2} \) |
| 59 | \( 1 - 2.52T + 59T^{2} \) |
| 61 | \( 1 + 8.35T + 61T^{2} \) |
| 67 | \( 1 + 5.98T + 67T^{2} \) |
| 71 | \( 1 - 8.76T + 71T^{2} \) |
| 73 | \( 1 - 10.7T + 73T^{2} \) |
| 79 | \( 1 + 9.64T + 79T^{2} \) |
| 83 | \( 1 - 15.6T + 83T^{2} \) |
| 89 | \( 1 + 3.36T + 89T^{2} \) |
| 97 | \( 1 - 0.313T + 97T^{2} \) |
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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
−8.522300117095898885108617793952, −7.78804179403441478674697943905, −7.58755449326336084881318289253, −6.68384512062370229043103065853, −5.75015372633790444626150422416, −4.36475657080634062962835699149, −3.45456382350992456091938117606, −2.41716613877441209929667129529, −1.90416128081973153514655828077, −0.979256416857560669136216252731,
0.979256416857560669136216252731, 1.90416128081973153514655828077, 2.41716613877441209929667129529, 3.45456382350992456091938117606, 4.36475657080634062962835699149, 5.75015372633790444626150422416, 6.68384512062370229043103065853, 7.58755449326336084881318289253, 7.78804179403441478674697943905, 8.522300117095898885108617793952
