| 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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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
−8.971834489648722980820436495974, −8.827237426648214521511927814205, −7.890841999667866464082158311270, −6.64742791763685139585923238575, −6.00982714490215939768839347134, −4.99782404197983560772229365488, −4.20803843465627590270069430813, −3.16349635156176399729002853251, −1.44706008845505384279685139626, 0,
1.44706008845505384279685139626, 3.16349635156176399729002853251, 4.20803843465627590270069430813, 4.99782404197983560772229365488, 6.00982714490215939768839347134, 6.64742791763685139585923238575, 7.890841999667866464082158311270, 8.827237426648214521511927814205, 8.971834489648722980820436495974
