| L(s) = 1 | + 3.79·3-s − 7.61·5-s + 7·7-s − 12.6·9-s − 11·11-s − 52.1·13-s − 28.8·15-s + 115.·17-s + 21.5·19-s + 26.5·21-s + 36.6·23-s − 66.9·25-s − 150.·27-s + 50.2·29-s − 153.·31-s − 41.7·33-s − 53.3·35-s + 303.·37-s − 197.·39-s + 301.·41-s + 273.·43-s + 96.1·45-s − 391.·47-s + 49·49-s + 439.·51-s + 119.·53-s + 83.7·55-s + ⋯ |
| L(s) = 1 | + 0.729·3-s − 0.681·5-s + 0.377·7-s − 0.467·9-s − 0.301·11-s − 1.11·13-s − 0.497·15-s + 1.65·17-s + 0.260·19-s + 0.275·21-s + 0.332·23-s − 0.535·25-s − 1.07·27-s + 0.321·29-s − 0.887·31-s − 0.220·33-s − 0.257·35-s + 1.34·37-s − 0.812·39-s + 1.14·41-s + 0.970·43-s + 0.318·45-s − 1.21·47-s + 0.142·49-s + 1.20·51-s + 0.308·53-s + 0.205·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.101560584\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.101560584\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| 11 | \( 1 + 11T \) |
| good | 3 | \( 1 - 3.79T + 27T^{2} \) |
| 5 | \( 1 + 7.61T + 125T^{2} \) |
| 13 | \( 1 + 52.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 115.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 21.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 36.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 50.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 153.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 303.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 301.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 273.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 391.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 119.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 243.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 824.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 310.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 705.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 331.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 812.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.37e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 604.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 963.T + 9.12e5T^{2} \) |
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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
−9.393428877509753637174760594886, −8.316306464403330074575821441435, −7.77952284582368230884626596482, −7.28690034104142681022543931708, −5.84672931427160532408678771216, −5.09217084864282387255551588473, −3.98190354729542565337931898118, −3.10495978109020133994994490383, −2.22780415439556682985418673262, −0.69876321166755813704292746145,
0.69876321166755813704292746145, 2.22780415439556682985418673262, 3.10495978109020133994994490383, 3.98190354729542565337931898118, 5.09217084864282387255551588473, 5.84672931427160532408678771216, 7.28690034104142681022543931708, 7.77952284582368230884626596482, 8.316306464403330074575821441435, 9.393428877509753637174760594886
