L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 4·7-s − 8-s + 9-s − 6·11-s + 12-s − 4·13-s + 4·14-s + 16-s − 6·17-s − 18-s + 8·19-s − 4·21-s + 6·22-s − 24-s − 5·25-s + 4·26-s + 27-s − 4·28-s + 8·31-s − 32-s − 6·33-s + 6·34-s + 36-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s − 1.80·11-s + 0.288·12-s − 1.10·13-s + 1.06·14-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 1.83·19-s − 0.872·21-s + 1.27·22-s − 0.204·24-s − 25-s + 0.784·26-s + 0.192·27-s − 0.755·28-s + 1.43·31-s − 0.176·32-s − 1.04·33-s + 1.02·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 438 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 438 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 73 | \( 1 - T \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−10.17016965724859756470772986706, −9.884788787844937526213219558952, −9.024516192101711121612799410469, −7.86531826005284337036409179905, −7.25763248586858216560634166408, −6.17568681060956116249671360793, −4.88158440838151026766227612735, −3.16405063493756711786253021465, −2.45895857818068025096950129576, 0,
2.45895857818068025096950129576, 3.16405063493756711786253021465, 4.88158440838151026766227612735, 6.17568681060956116249671360793, 7.25763248586858216560634166408, 7.86531826005284337036409179905, 9.024516192101711121612799410469, 9.884788787844937526213219558952, 10.17016965724859756470772986706