L(s) = 1 | + 3-s + 2.23·5-s + 4.61·7-s + 9-s + 2.23·11-s − 1.76·13-s + 2.23·15-s − 4.85·17-s + 8.09·19-s + 4.61·21-s + 2.38·23-s + 27-s + 8.61·29-s − 9.56·31-s + 2.23·33-s + 10.3·35-s − 6.85·37-s − 1.76·39-s − 3.09·41-s − 4.70·43-s + 2.23·45-s + 4.14·47-s + 14.3·49-s − 4.85·51-s + 1.76·53-s + 5.00·55-s + 8.09·57-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.999·5-s + 1.74·7-s + 0.333·9-s + 0.674·11-s − 0.489·13-s + 0.577·15-s − 1.17·17-s + 1.85·19-s + 1.00·21-s + 0.496·23-s + 0.192·27-s + 1.60·29-s − 1.71·31-s + 0.389·33-s + 1.74·35-s − 1.12·37-s − 0.282·39-s − 0.482·41-s − 0.717·43-s + 0.333·45-s + 0.604·47-s + 2.04·49-s − 0.679·51-s + 0.242·53-s + 0.674·55-s + 1.07·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2832 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.575944559\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.575944559\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 59 | \( 1 - T \) |
good | 5 | \( 1 - 2.23T + 5T^{2} \) |
| 7 | \( 1 - 4.61T + 7T^{2} \) |
| 11 | \( 1 - 2.23T + 11T^{2} \) |
| 13 | \( 1 + 1.76T + 13T^{2} \) |
| 17 | \( 1 + 4.85T + 17T^{2} \) |
| 19 | \( 1 - 8.09T + 19T^{2} \) |
| 23 | \( 1 - 2.38T + 23T^{2} \) |
| 29 | \( 1 - 8.61T + 29T^{2} \) |
| 31 | \( 1 + 9.56T + 31T^{2} \) |
| 37 | \( 1 + 6.85T + 37T^{2} \) |
| 41 | \( 1 + 3.09T + 41T^{2} \) |
| 43 | \( 1 + 4.70T + 43T^{2} \) |
| 47 | \( 1 - 4.14T + 47T^{2} \) |
| 53 | \( 1 - 1.76T + 53T^{2} \) |
| 61 | \( 1 + 9.85T + 61T^{2} \) |
| 67 | \( 1 - 2.70T + 67T^{2} \) |
| 71 | \( 1 + 9.94T + 71T^{2} \) |
| 73 | \( 1 + 5.85T + 73T^{2} \) |
| 79 | \( 1 - 3T + 79T^{2} \) |
| 83 | \( 1 + 0.618T + 83T^{2} \) |
| 89 | \( 1 - 10.7T + 89T^{2} \) |
| 97 | \( 1 - 3T + 97T^{2} \) |
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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
−8.968203753221773784895668290035, −8.077094650274238793911463423579, −7.33462980900433125081783652752, −6.66607602280328326536838434231, −5.45895147662890354971884211304, −5.01155601760741175285584453177, −4.13137093560284460236042063632, −2.94193626494116519172233129288, −1.92661075176228879542443297644, −1.35277144872317096191075365986,
1.35277144872317096191075365986, 1.92661075176228879542443297644, 2.94193626494116519172233129288, 4.13137093560284460236042063632, 5.01155601760741175285584453177, 5.45895147662890354971884211304, 6.66607602280328326536838434231, 7.33462980900433125081783652752, 8.077094650274238793911463423579, 8.968203753221773784895668290035