L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 7-s − 8-s + 9-s + 10-s + 3·11-s − 12-s + 13-s + 14-s + 15-s + 16-s + 6·17-s − 18-s − 2·19-s − 20-s + 21-s − 3·22-s + 2·23-s + 24-s + 25-s − 26-s − 27-s − 28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.904·11-s − 0.288·12-s + 0.277·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s + 1.45·17-s − 0.235·18-s − 0.458·19-s − 0.223·20-s + 0.218·21-s − 0.639·22-s + 0.417·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 \) |
good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−12.46900771108099, −12.15602045221746, −11.88203185035956, −11.31440602503883, −10.82594298388712, −10.55110916170271, −9.951196562624536, −9.574130564786320, −9.212332188984496, −8.510933263592379, −8.231459711854921, −7.757730330971419, −7.067893674827641, −6.853128591521262, −6.235262167154593, −6.002343639011869, −5.252519170438860, −4.810995827970506, −4.197765608181780, −3.579741603045718, −3.216701606045196, −2.634711700165313, −1.701936491890810, −1.284912023468405, −0.7261202088611445, 0,
0.7261202088611445, 1.284912023468405, 1.701936491890810, 2.634711700165313, 3.216701606045196, 3.579741603045718, 4.197765608181780, 4.810995827970506, 5.252519170438860, 6.002343639011869, 6.235262167154593, 6.853128591521262, 7.067893674827641, 7.757730330971419, 8.231459711854921, 8.510933263592379, 9.212332188984496, 9.574130564786320, 9.951196562624536, 10.55110916170271, 10.82594298388712, 11.31440602503883, 11.88203185035956, 12.15602045221746, 12.46900771108099