| L(s) = 1 | − 3.00·2-s + 1.04·4-s + 20.8·5-s − 18.7·7-s + 20.9·8-s − 62.7·10-s + 3.28·11-s − 48.5·13-s + 56.2·14-s − 71.2·16-s + 62.6·17-s − 18.9·19-s + 21.8·20-s − 9.88·22-s + 53.1·23-s + 309.·25-s + 146.·26-s − 19.6·28-s − 200.·29-s − 125.·31-s + 47.1·32-s − 188.·34-s − 389.·35-s − 190.·37-s + 57.0·38-s + 435.·40-s + 273.·41-s + ⋯ |
| L(s) = 1 | − 1.06·2-s + 0.131·4-s + 1.86·5-s − 1.00·7-s + 0.924·8-s − 1.98·10-s + 0.0900·11-s − 1.03·13-s + 1.07·14-s − 1.11·16-s + 0.893·17-s − 0.229·19-s + 0.244·20-s − 0.0957·22-s + 0.482·23-s + 2.47·25-s + 1.10·26-s − 0.132·28-s − 1.28·29-s − 0.725·31-s + 0.260·32-s − 0.950·34-s − 1.88·35-s − 0.844·37-s + 0.243·38-s + 1.72·40-s + 1.04·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.254178977\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.254178977\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 - 239T \) |
| good | 2 | \( 1 + 3.00T + 8T^{2} \) |
| 5 | \( 1 - 20.8T + 125T^{2} \) |
| 7 | \( 1 + 18.7T + 343T^{2} \) |
| 11 | \( 1 - 3.28T + 1.33e3T^{2} \) |
| 13 | \( 1 + 48.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 62.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 18.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 53.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 200.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 125.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 190.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 273.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 101.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 497.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 377.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 455.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 501.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 680.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 572.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 103.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 73.2T + 4.93e5T^{2} \) |
| 83 | \( 1 - 551.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 333.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 206.T + 9.12e5T^{2} \) |
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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
−9.185740570563843069494511069762, −8.120343236789219827704656187317, −7.11487352325764400207507689871, −6.59756380920978251947439477763, −5.54625144396981469140823147234, −5.06699052207682115405959658469, −3.61866464536899650156403079493, −2.45481977659732889369312448473, −1.71119534703919822493652406223, −0.60217368110743135476728093528,
0.60217368110743135476728093528, 1.71119534703919822493652406223, 2.45481977659732889369312448473, 3.61866464536899650156403079493, 5.06699052207682115405959658469, 5.54625144396981469140823147234, 6.59756380920978251947439477763, 7.11487352325764400207507689871, 8.120343236789219827704656187317, 9.185740570563843069494511069762
