L(s) = 1 | + 5-s + 11-s + 13-s − 7·17-s + 4·19-s − 3·23-s + 25-s − 2·29-s + 6·31-s − 9·37-s + 9·41-s + 2·43-s − 2·47-s + 6·53-s + 55-s + 5·59-s − 2·61-s + 65-s − 5·67-s − 8·71-s − 7·73-s − 79-s + 14·83-s − 7·85-s + 7·89-s + 4·95-s − 97-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 0.301·11-s + 0.277·13-s − 1.69·17-s + 0.917·19-s − 0.625·23-s + 1/5·25-s − 0.371·29-s + 1.07·31-s − 1.47·37-s + 1.40·41-s + 0.304·43-s − 0.291·47-s + 0.824·53-s + 0.134·55-s + 0.650·59-s − 0.256·61-s + 0.124·65-s − 0.610·67-s − 0.949·71-s − 0.819·73-s − 0.112·79-s + 1.53·83-s − 0.759·85-s + 0.741·89-s + 0.410·95-s − 0.101·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 11 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−13.30203861592243, −12.74449668735043, −12.18655064467063, −11.74469644527653, −11.37835060053214, −10.78949575851151, −10.39379349170635, −9.947776417790588, −9.290187329223837, −9.040825382828225, −8.563828557725401, −8.029629850262542, −7.355097585673546, −7.051753916883821, −6.288704086314898, −6.185024713823961, −5.437671055294540, −4.982909979700053, −4.327539992386368, −3.972576054380659, −3.269404595564418, −2.637617559957270, −2.134731414930609, −1.524525070610822, −0.8365148932946873, 0,
0.8365148932946873, 1.524525070610822, 2.134731414930609, 2.637617559957270, 3.269404595564418, 3.972576054380659, 4.327539992386368, 4.982909979700053, 5.437671055294540, 6.185024713823961, 6.288704086314898, 7.051753916883821, 7.355097585673546, 8.029629850262542, 8.563828557725401, 9.040825382828225, 9.290187329223837, 9.947776417790588, 10.39379349170635, 10.78949575851151, 11.37835060053214, 11.74469644527653, 12.18655064467063, 12.74449668735043, 13.30203861592243