L(s) = 1 | − 3-s − 7-s + 9-s + 11-s + 13-s − 7·17-s + 21-s − 27-s − 5·29-s − 31-s − 33-s + 8·37-s − 39-s + 6·41-s + 8·43-s − 5·47-s − 6·49-s + 7·51-s + 53-s + 3·59-s + 7·61-s − 63-s + 7·67-s − 12·73-s − 77-s − 12·79-s + 81-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.301·11-s + 0.277·13-s − 1.69·17-s + 0.218·21-s − 0.192·27-s − 0.928·29-s − 0.179·31-s − 0.174·33-s + 1.31·37-s − 0.160·39-s + 0.937·41-s + 1.21·43-s − 0.729·47-s − 6/7·49-s + 0.980·51-s + 0.137·53-s + 0.390·59-s + 0.896·61-s − 0.125·63-s + 0.855·67-s − 1.40·73-s − 0.113·77-s − 1.35·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 5 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 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
−14.66729327870449, −14.00798554659308, −13.27972045439504, −12.93315887388148, −12.77016897067093, −11.80741283218511, −11.47915820283730, −11.01099343110011, −10.64649311539283, −9.830195876990443, −9.477041411093381, −8.935890280779203, −8.410126447224952, −7.688914812482813, −7.134410061828126, −6.628592128934096, −6.102538095192961, −5.684003925785379, −4.935368780475390, −4.219489716667828, −4.013958281542822, −3.055243617849134, −2.385545043233777, −1.698044082115498, −0.8068954370116120, 0,
0.8068954370116120, 1.698044082115498, 2.385545043233777, 3.055243617849134, 4.013958281542822, 4.219489716667828, 4.935368780475390, 5.684003925785379, 6.102538095192961, 6.628592128934096, 7.134410061828126, 7.688914812482813, 8.410126447224952, 8.935890280779203, 9.477041411093381, 9.830195876990443, 10.64649311539283, 11.01099343110011, 11.47915820283730, 11.80741283218511, 12.77016897067093, 12.93315887388148, 13.27972045439504, 14.00798554659308, 14.66729327870449