| L(s) = 1 | + 0.381·2-s + 0.874·3-s − 1.85·4-s + 0.333·6-s − 1.47·8-s − 2.23·9-s + 2.23·11-s − 1.62·12-s − 4.03·13-s + 3.14·16-s + 7.40·17-s − 0.854·18-s + 4.24·19-s + 0.854·22-s + 3.76·23-s − 1.28·24-s − 1.54·26-s − 4.57·27-s + 2.23·29-s + 6.86·31-s + 4.14·32-s + 1.95·33-s + 2.82·34-s + 4.14·36-s + 10.7·37-s + 1.62·38-s − 3.52·39-s + ⋯ |
| L(s) = 1 | + 0.270·2-s + 0.504·3-s − 0.927·4-s + 0.136·6-s − 0.520·8-s − 0.745·9-s + 0.674·11-s − 0.467·12-s − 1.11·13-s + 0.786·16-s + 1.79·17-s − 0.201·18-s + 0.973·19-s + 0.182·22-s + 0.784·23-s − 0.262·24-s − 0.302·26-s − 0.880·27-s + 0.415·29-s + 1.23·31-s + 0.732·32-s + 0.340·33-s + 0.485·34-s + 0.690·36-s + 1.76·37-s + 0.262·38-s − 0.564·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.723512286\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.723512286\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 0.381T + 2T^{2} \) |
| 3 | \( 1 - 0.874T + 3T^{2} \) |
| 11 | \( 1 - 2.23T + 11T^{2} \) |
| 13 | \( 1 + 4.03T + 13T^{2} \) |
| 17 | \( 1 - 7.40T + 17T^{2} \) |
| 19 | \( 1 - 4.24T + 19T^{2} \) |
| 23 | \( 1 - 3.76T + 23T^{2} \) |
| 29 | \( 1 - 2.23T + 29T^{2} \) |
| 31 | \( 1 - 6.86T + 31T^{2} \) |
| 37 | \( 1 - 10.7T + 37T^{2} \) |
| 41 | \( 1 + 4.78T + 41T^{2} \) |
| 43 | \( 1 - 5T + 43T^{2} \) |
| 47 | \( 1 + 9.48T + 47T^{2} \) |
| 53 | \( 1 - 9.70T + 53T^{2} \) |
| 59 | \( 1 + 13.1T + 59T^{2} \) |
| 61 | \( 1 + 3.03T + 61T^{2} \) |
| 67 | \( 1 - 8.70T + 67T^{2} \) |
| 71 | \( 1 + 7.47T + 71T^{2} \) |
| 73 | \( 1 - 2.62T + 73T^{2} \) |
| 79 | \( 1 + 4.70T + 79T^{2} \) |
| 83 | \( 1 - 6.86T + 83T^{2} \) |
| 89 | \( 1 - 7.94T + 89T^{2} \) |
| 97 | \( 1 + 10.1T + 97T^{2} \) |
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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
−9.621240875598589495070815766931, −9.018522445632058270191866144420, −8.080359635468571264499353096271, −7.53562621692909125231051199261, −6.21653577053897407005030519307, −5.35456083259767857676768520788, −4.61772433744267444750778081059, −3.43927221409349693872136768565, −2.81326485793824964433691962308, −0.955988543724692389751162287094,
0.955988543724692389751162287094, 2.81326485793824964433691962308, 3.43927221409349693872136768565, 4.61772433744267444750778081059, 5.35456083259767857676768520788, 6.21653577053897407005030519307, 7.53562621692909125231051199261, 8.080359635468571264499353096271, 9.018522445632058270191866144420, 9.621240875598589495070815766931
