| L(s) = 1 | − 7-s − 3·9-s − 11-s − 13-s − 5·19-s − 5·29-s + 2·31-s − 4·37-s − 5·41-s + 9·43-s − 6·47-s − 6·49-s + 2·53-s − 8·59-s + 8·61-s + 3·63-s − 8·67-s + 10·71-s + 3·73-s + 77-s − 3·79-s + 9·81-s − 3·83-s − 10·89-s + 91-s − 2·97-s + 3·99-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 9-s − 0.301·11-s − 0.277·13-s − 1.14·19-s − 0.928·29-s + 0.359·31-s − 0.657·37-s − 0.780·41-s + 1.37·43-s − 0.875·47-s − 6/7·49-s + 0.274·53-s − 1.04·59-s + 1.02·61-s + 0.377·63-s − 0.977·67-s + 1.18·71-s + 0.351·73-s + 0.113·77-s − 0.337·79-s + 81-s − 0.329·83-s − 1.05·89-s + 0.104·91-s − 0.203·97-s + 0.301·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 211600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 211600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.24213886110817, −12.72506834467972, −12.39674368693017, −11.83780307055722, −11.30686290064377, −10.97903800565240, −10.46141701980604, −9.984636602165827, −9.452660338151083, −8.999265062556307, −8.478723930796937, −8.145805353982929, −7.540183220183239, −7.037695333060061, −6.370945648504177, −6.151634162401228, −5.431848780609729, −5.101323888356897, −4.411305136047033, −3.864158118323200, −3.239594729593786, −2.789540649366389, −2.157953154324139, −1.636340283496817, −0.5765472876696678, 0,
0.5765472876696678, 1.636340283496817, 2.157953154324139, 2.789540649366389, 3.239594729593786, 3.864158118323200, 4.411305136047033, 5.101323888356897, 5.431848780609729, 6.151634162401228, 6.370945648504177, 7.037695333060061, 7.540183220183239, 8.145805353982929, 8.478723930796937, 8.999265062556307, 9.452660338151083, 9.984636602165827, 10.46141701980604, 10.97903800565240, 11.30686290064377, 11.83780307055722, 12.39674368693017, 12.72506834467972, 13.24213886110817
