L(s) = 1 | − 2·4-s − 3·5-s − 7-s − 3·11-s + 4·13-s + 4·16-s + 3·17-s + 6·20-s + 4·25-s + 2·28-s + 6·29-s + 4·31-s + 3·35-s − 2·37-s − 6·41-s − 43-s + 6·44-s + 3·47-s − 6·49-s − 8·52-s + 12·53-s + 9·55-s − 6·59-s − 61-s − 8·64-s − 12·65-s + 4·67-s + ⋯ |
L(s) = 1 | − 4-s − 1.34·5-s − 0.377·7-s − 0.904·11-s + 1.10·13-s + 16-s + 0.727·17-s + 1.34·20-s + 4/5·25-s + 0.377·28-s + 1.11·29-s + 0.718·31-s + 0.507·35-s − 0.328·37-s − 0.937·41-s − 0.152·43-s + 0.904·44-s + 0.437·47-s − 6/7·49-s − 1.10·52-s + 1.64·53-s + 1.21·55-s − 0.781·59-s − 0.128·61-s − 64-s − 1.48·65-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−8.326953329257171788522384866930, −7.78372256126247095407355625797, −6.89978211559514220466164652247, −5.90249649788673780822780667845, −5.09166130381690209084265412383, −4.29625259461062490239638954547, −3.60292617324638028380948388621, −2.93315543379839730134263284965, −1.08948667272131778507674861959, 0,
1.08948667272131778507674861959, 2.93315543379839730134263284965, 3.60292617324638028380948388621, 4.29625259461062490239638954547, 5.09166130381690209084265412383, 5.90249649788673780822780667845, 6.89978211559514220466164652247, 7.78372256126247095407355625797, 8.326953329257171788522384866930