L(s) = 1 | − 3-s + 3·7-s − 2·9-s − 4·11-s + 3·13-s + 3·17-s − 3·21-s − 23-s − 5·25-s + 5·27-s − 29-s + 4·31-s + 4·33-s − 6·37-s − 3·39-s − 4·41-s + 4·47-s + 2·49-s − 3·51-s − 53-s − 9·59-s + 8·61-s − 6·63-s + 3·67-s + 69-s − 12·71-s − 3·73-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.13·7-s − 2/3·9-s − 1.20·11-s + 0.832·13-s + 0.727·17-s − 0.654·21-s − 0.208·23-s − 25-s + 0.962·27-s − 0.185·29-s + 0.718·31-s + 0.696·33-s − 0.986·37-s − 0.480·39-s − 0.624·41-s + 0.583·47-s + 2/7·49-s − 0.420·51-s − 0.137·53-s − 1.17·59-s + 1.02·61-s − 0.755·63-s + 0.366·67-s + 0.120·69-s − 1.42·71-s − 0.351·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−14.08869088283544, −13.66412110073179, −13.13235287185972, −12.54150336978921, −11.96390430772857, −11.56847973985588, −11.24345583577766, −10.63427847729762, −10.28691166969808, −9.812822393201625, −8.895081148535867, −8.491704288587471, −8.125131018875121, −7.570226814345819, −7.128176817306228, −6.129591009787267, −5.933042776087916, −5.338750903910377, −4.902737901284888, −4.372504935685676, −3.510771138339633, −3.037723308244794, −2.207416497637545, −1.639377636770503, −0.8246611440423830, 0,
0.8246611440423830, 1.639377636770503, 2.207416497637545, 3.037723308244794, 3.510771138339633, 4.372504935685676, 4.902737901284888, 5.338750903910377, 5.933042776087916, 6.129591009787267, 7.128176817306228, 7.570226814345819, 8.125131018875121, 8.491704288587471, 8.895081148535867, 9.812822393201625, 10.28691166969808, 10.63427847729762, 11.24345583577766, 11.56847973985588, 11.96390430772857, 12.54150336978921, 13.13235287185972, 13.66412110073179, 14.08869088283544