L(s) = 1 | − 3-s − 5-s − 7-s − 2·9-s + 13-s + 15-s − 3·17-s − 19-s + 21-s + 3·23-s + 25-s + 5·27-s + 3·29-s + 2·31-s + 35-s + 10·37-s − 39-s + 6·41-s − 2·43-s + 2·45-s − 6·49-s + 3·51-s − 3·53-s + 57-s − 3·59-s − 8·61-s + 2·63-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s − 2/3·9-s + 0.277·13-s + 0.258·15-s − 0.727·17-s − 0.229·19-s + 0.218·21-s + 0.625·23-s + 1/5·25-s + 0.962·27-s + 0.557·29-s + 0.359·31-s + 0.169·35-s + 1.64·37-s − 0.160·39-s + 0.937·41-s − 0.304·43-s + 0.298·45-s − 6/7·49-s + 0.420·51-s − 0.412·53-s + 0.132·57-s − 0.390·59-s − 1.02·61-s + 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−7.83888869496288642956438188125, −6.74852187375451275996803010992, −6.40463811695184100892327804488, −5.63428152509819530143446151281, −4.80490941937650326164608414961, −4.16656717182338968423331235359, −3.15423951617049884044133077897, −2.48743934183121311255565093589, −1.06274698769762861405542526326, 0,
1.06274698769762861405542526326, 2.48743934183121311255565093589, 3.15423951617049884044133077897, 4.16656717182338968423331235359, 4.80490941937650326164608414961, 5.63428152509819530143446151281, 6.40463811695184100892327804488, 6.74852187375451275996803010992, 7.83888869496288642956438188125