L(s) = 1 | − 2-s + 4-s + 5-s + 2·7-s − 8-s − 10-s + 5·11-s − 2·14-s + 16-s − 5·17-s − 5·19-s + 20-s − 5·22-s + 4·23-s + 25-s + 2·28-s − 4·29-s + 2·31-s − 32-s + 5·34-s + 2·35-s − 2·37-s + 5·38-s − 40-s + 3·41-s + 5·44-s − 4·46-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.755·7-s − 0.353·8-s − 0.316·10-s + 1.50·11-s − 0.534·14-s + 1/4·16-s − 1.21·17-s − 1.14·19-s + 0.223·20-s − 1.06·22-s + 0.834·23-s + 1/5·25-s + 0.377·28-s − 0.742·29-s + 0.359·31-s − 0.176·32-s + 0.857·34-s + 0.338·35-s − 0.328·37-s + 0.811·38-s − 0.158·40-s + 0.468·41-s + 0.753·44-s − 0.589·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 166410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 166410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 43 | \( 1 \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 5 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−13.58241235674279, −12.92950916455115, −12.47688267020736, −11.99258247490945, −11.34159514993027, −11.04348887825486, −10.84045258344744, −10.05546914490758, −9.520506829705483, −9.200482733217530, −8.640112190917865, −8.428120305377129, −7.745028309069952, −7.120977003288016, −6.559723852670877, −6.447656861193186, −5.747430500460578, −4.967761555111092, −4.634145144224944, −3.874398325796253, −3.501834725209928, −2.382803934530585, −2.194257299176380, −1.456091250834658, −0.9561822728934380, 0,
0.9561822728934380, 1.456091250834658, 2.194257299176380, 2.382803934530585, 3.501834725209928, 3.874398325796253, 4.634145144224944, 4.967761555111092, 5.747430500460578, 6.447656861193186, 6.559723852670877, 7.120977003288016, 7.745028309069952, 8.428120305377129, 8.640112190917865, 9.200482733217530, 9.520506829705483, 10.05546914490758, 10.84045258344744, 11.04348887825486, 11.34159514993027, 11.99258247490945, 12.47688267020736, 12.92950916455115, 13.58241235674279