| L(s) = 1 | + 2.23i·5-s + 3.01i·7-s + 4.03i·11-s − 1.33·13-s + 11.3i·17-s − 18.6i·19-s + (−22.9 + 1.15i)23-s − 5.00·25-s + 40.0·29-s − 37.8·31-s − 6.74·35-s + 3.61i·37-s − 62.5·41-s − 62.6i·43-s − 7.72·47-s + ⋯ |
| L(s) = 1 | + 0.447i·5-s + 0.430i·7-s + 0.366i·11-s − 0.102·13-s + 0.669i·17-s − 0.982i·19-s + (−0.998 + 0.0501i)23-s − 0.200·25-s + 1.38·29-s − 1.22·31-s − 0.192·35-s + 0.0978i·37-s − 1.52·41-s − 1.45i·43-s − 0.164·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0501 + 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0501 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8755664697\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8755664697\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 2.23iT \) |
| 23 | \( 1 + (22.9 - 1.15i)T \) |
| good | 7 | \( 1 - 3.01iT - 49T^{2} \) |
| 11 | \( 1 - 4.03iT - 121T^{2} \) |
| 13 | \( 1 + 1.33T + 169T^{2} \) |
| 17 | \( 1 - 11.3iT - 289T^{2} \) |
| 19 | \( 1 + 18.6iT - 361T^{2} \) |
| 29 | \( 1 - 40.0T + 841T^{2} \) |
| 31 | \( 1 + 37.8T + 961T^{2} \) |
| 37 | \( 1 - 3.61iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 62.5T + 1.68e3T^{2} \) |
| 43 | \( 1 + 62.6iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 7.72T + 2.20e3T^{2} \) |
| 53 | \( 1 - 41.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 12.9T + 3.48e3T^{2} \) |
| 61 | \( 1 + 43.9iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 65.9iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 33.0T + 5.04e3T^{2} \) |
| 73 | \( 1 + 33.5T + 5.32e3T^{2} \) |
| 79 | \( 1 - 88.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 102. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 150. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 35.2iT - 9.40e3T^{2} \) |
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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
−8.140219831462936391946723356956, −7.22963894151432251569344358210, −6.68625466858142597510426415977, −5.87814735822563130626799036390, −5.13005927226063117261087630982, −4.25650689072774018852450631946, −3.40177523481525900048809590725, −2.47570590483563097866688282284, −1.68990302856884444711104305529, −0.19984436334406190478650055964,
0.933746362815750867893924103188, 1.91580983865110248862994659984, 3.05137730707319485107573716554, 3.88843658879807007777242449807, 4.65060983564916510264443620231, 5.46618202130107539996682727583, 6.18145918720937944076998790687, 7.00992193585685160247174198388, 7.76552053883495904167766758443, 8.397150911366318934294810265074
