| L(s) = 1 | − 3-s − 3·5-s + 9-s − 3·11-s + 5·13-s + 3·15-s + 17-s + 2·23-s + 4·25-s − 27-s + 4·29-s − 6·31-s + 3·33-s − 11·37-s − 5·39-s + 2·41-s + 5·43-s − 3·45-s − 10·47-s − 51-s + 53-s + 9·55-s + 12·59-s + 2·61-s − 15·65-s − 13·67-s − 2·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s + 1/3·9-s − 0.904·11-s + 1.38·13-s + 0.774·15-s + 0.242·17-s + 0.417·23-s + 4/5·25-s − 0.192·27-s + 0.742·29-s − 1.07·31-s + 0.522·33-s − 1.80·37-s − 0.800·39-s + 0.312·41-s + 0.762·43-s − 0.447·45-s − 1.45·47-s − 0.140·51-s + 0.137·53-s + 1.21·55-s + 1.56·59-s + 0.256·61-s − 1.86·65-s − 1.58·67-s − 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−13.44946836612263, −12.88364601560288, −12.55522166501459, −12.07660691876240, −11.52726947282425, −11.14305215286464, −10.85158322555724, −10.30870042645129, −9.873891546346802, −9.003888625794130, −8.657454483438285, −8.160530907891061, −7.728253018726293, −7.192570569336929, −6.760738470800787, −6.162367712185252, −5.536954613672452, −5.121992394202605, −4.564009640458964, −3.881815656153984, −3.544610357268166, −3.026675313301776, −2.147467914307510, −1.366571826567997, −0.6758690858641896, 0,
0.6758690858641896, 1.366571826567997, 2.147467914307510, 3.026675313301776, 3.544610357268166, 3.881815656153984, 4.564009640458964, 5.121992394202605, 5.536954613672452, 6.162367712185252, 6.760738470800787, 7.192570569336929, 7.728253018726293, 8.160530907891061, 8.657454483438285, 9.003888625794130, 9.873891546346802, 10.30870042645129, 10.85158322555724, 11.14305215286464, 11.52726947282425, 12.07660691876240, 12.55522166501459, 12.88364601560288, 13.44946836612263
