| L(s) = 1 | − 2-s − 2.22·3-s + 4-s + 3.08·5-s + 2.22·6-s + 7-s − 8-s + 1.96·9-s − 3.08·10-s + 1.14·11-s − 2.22·12-s + 3.25·13-s − 14-s − 6.87·15-s + 16-s + 4.21·17-s − 1.96·18-s + 3.08·20-s − 2.22·21-s − 1.14·22-s − 0.445·23-s + 2.22·24-s + 4.52·25-s − 3.25·26-s + 2.30·27-s + 28-s + 9.88·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.28·3-s + 0.5·4-s + 1.38·5-s + 0.909·6-s + 0.377·7-s − 0.353·8-s + 0.654·9-s − 0.976·10-s + 0.345·11-s − 0.643·12-s + 0.903·13-s − 0.267·14-s − 1.77·15-s + 0.250·16-s + 1.02·17-s − 0.462·18-s + 0.690·20-s − 0.486·21-s − 0.244·22-s − 0.0929·23-s + 0.454·24-s + 0.905·25-s − 0.639·26-s + 0.444·27-s + 0.188·28-s + 1.83·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.425919821\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.425919821\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 2.22T + 3T^{2} \) |
| 5 | \( 1 - 3.08T + 5T^{2} \) |
| 11 | \( 1 - 1.14T + 11T^{2} \) |
| 13 | \( 1 - 3.25T + 13T^{2} \) |
| 17 | \( 1 - 4.21T + 17T^{2} \) |
| 23 | \( 1 + 0.445T + 23T^{2} \) |
| 29 | \( 1 - 9.88T + 29T^{2} \) |
| 31 | \( 1 + 1.67T + 31T^{2} \) |
| 37 | \( 1 + 8.76T + 37T^{2} \) |
| 41 | \( 1 + 4.91T + 41T^{2} \) |
| 43 | \( 1 - 4.44T + 43T^{2} \) |
| 47 | \( 1 - 2.20T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 - 6.58T + 59T^{2} \) |
| 61 | \( 1 - 7.80T + 61T^{2} \) |
| 67 | \( 1 - 0.524T + 67T^{2} \) |
| 71 | \( 1 + 4.81T + 71T^{2} \) |
| 73 | \( 1 - 8.13T + 73T^{2} \) |
| 79 | \( 1 + 10.7T + 79T^{2} \) |
| 83 | \( 1 + 13.1T + 83T^{2} \) |
| 89 | \( 1 - 16.8T + 89T^{2} \) |
| 97 | \( 1 + 13.5T + 97T^{2} \) |
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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
−8.556824217909463704006292193526, −7.34434110807010958667164701513, −6.63112625855775960052118159272, −6.07446068325385972313322481977, −5.52877595008814571886929260656, −4.95340597976309571238324319024, −3.71672106357049880207420091488, −2.54633872661957810196861549986, −1.50213481403693836158447806388, −0.861434519658450304312551620221,
0.861434519658450304312551620221, 1.50213481403693836158447806388, 2.54633872661957810196861549986, 3.71672106357049880207420091488, 4.95340597976309571238324319024, 5.52877595008814571886929260656, 6.07446068325385972313322481977, 6.63112625855775960052118159272, 7.34434110807010958667164701513, 8.556824217909463704006292193526
